Analysis of failure in timber boards under tensile loading initiated by knots

Analysis of failure in timber boards under tensile loading initiated
by knots-a study of basic failure mechanism
Project Work
Approved by the Faculty of Civil Engineering
of the Technische Universität Dresden
Written by
Mohhammad Afsar Sujon
Supervisors: Univ. - Prof. Dr.-Ing. habil. Michael Kaliske
Scientific Consultant. Dipl.-Ing. Christian Jenkel
Date of submission: 27-03-2015
Date of presentation: 29-04-2015

Acknowledgments
Thinking in retrospective and recalling the time when I was given this project work in the
Fracture mechanics models of timber, I am thankful for the help received to overcome such
challenge, provided by the project coordinator, Dipl.-Ing. Christian Jenkel. I highly recognize
his sincere guidance. I also would like to thank him for the guidance to become familiar with
the basic problem of timber with knots. I am also thankful to him for his help to learn the
software SIMULIA ABAQUS which needed to simulate timber behavior under tension.
At the end, I shall not forget to thank all the teachers who throughout the years taught the
science of Engineering, enabling me to complete my project.
Dresden, 27.03.2015 Mohhammad Afsar Sujon

Declaration of independent work
I confirm that this assignment is my own work and that I have not sought or used unacceptable
help of third parties to produce this work and that I have clearly referenced all sources used in
the work. I have fully referenced and used inverted commas for all text directly or indirectly
quoted from a source.
This work has not yet been submitted to another examination institution – neither in Germany
nor outside Germany – neither in the same nor in a similar way and has not yet been published.
Dresden,
……………………………………………
(Signature)

Abstract
Wood can be characterized as a natural, cellular, polymer-based, hygrothermal viscoelastic
material. As a construction material, it has been used very early next to stone, owing to its good
material and mechanical properties. It can be fabricated to a variety of shapes and sizes; and
not the least important- economically available. Wood is a renewable and biodegradable
resource. Its main drawbacks are: wood is an anisotropic material with an array of defects in
the form of irregular grains and knots; it is subject to decay if not kept dry, and it is flammable.
In the last four decades, the finite element method, FEM, has become the prevalent technique
used for analyzing physical phenomena in the field of structural, solid and fluid mechanics as
well as for the solution of field problems. This project paper gives a review of the published
papers dealing with finite element methods applied to wood. Especially the paper related to
tension parallel to the grain and also with the knot problems.
Wood is a material with a microstructure reflected on the macro scale in its grain. Cell walls
are layered and contain three organic components: cellulose, hemicellulose and natures
adhesive, lignin. The lay-up of cellulose fibers in the wall is complex but important because it
accounts for the part of the great anisotropy of wood. Micro and macro structure of wood are
analyzed in this project.
Studies of wood from a micro to a macro level are necessary for a more precise definition and
understanding of material and mechanical behavior of wood. At the micro level, the fiber
shape, cell wall thickness, etc. are included in modeling. Their continuum properties can be
derived by use of a homogenization procedure and the finite element method.
Wood is regarded as a brittle material, depending on stress direction, duration of loads and the
moisture. Different wood species of softwoods as well as hard woods due to the orthotropic
nature have been studied in different publication by finite elements in various crack
propagation systems. These results have shown that softwoods and hardwoods are quite
different in their micro structures.
Wood under static or quasi-static loads is used for example in trusses, buildings, bridges and
other important structures. So, it is very important to know the wood properties under different
load condition to ensure the safety of the structures by increasing it structural stability. The
research can also widen the sector of using timber by utilizing its tension strength properties.
This project mainly concerns theoretical work regarding parallel to grain fracture in wooden
structural elements. It focused on different models for strength and fracture analysis, based on
fracture mechanics approaches, and their application to analysis of timber boards with a knot
loaded parallel to grain.

Analysis of failure in timber boards under tensile loading initiated by knots [5]
Table of contents
Chapter 1....................................................................................................................................7
Introduction................................................................................................................................7
1.1 General remarks ...............................................................................................................7
1.2 Objectives.........................................................................................................................8
1.3 Organization of the Project ..............................................................................................9
Chapter 2..................................................................................................................................10
Properties of Timber ................................................................................................................10
2.1 Macrostructure and Microstructure of Wood.................................................................10
2.2 Physical Properties of Wood..........................................................................................16
2.3 Mechanical Properties of Timber...................................................................................18
2.4 Failure types...................................................................................................................23
2.5 Strength, toughness, failure and fracture morphology...................................................32
Chapter 3..................................................................................................................................37
Basic Fracture Mechanics and Modelling Of Timber..............................................................37
3.1 Introduction of the Fracture Mechanics .........................................................................37
3.2 Fracture mechanics models............................................................................................38
3.2.1 Linear elastic fracture mechanics models ...................................................................38
3.2.2 Non-linear elastic fracture models ..............................................................................39
Fictitious crack model, FCM................................................................................................40
Bridged crack model ............................................................................................................40
Lattice fracture model ..........................................................................................................42
3.3 Modelling of timber properties ......................................................................................44
3.4 FEM at large deformations and brittle failure prediction...............................................48
Chapter 4..................................................................................................................................49
Effect of Knots in Timber........................................................................................................49
4.1 Knots ..............................................................................................................................49
4.2 Investigation of the Deformation Behavior....................................................................50
4.3 Interaction of Knots........................................................................................................55
4.4 Strain History up to Failure............................................................................................58
4.5 Correlation between Strains and Failure Behavior ........................................................59
Chapter 5..................................................................................................................................62

Finite Element Modelling of Timber Boards with Knots........................................................62
5.1 Cohesive zone Model.....................................................................................................63
5.1.1 Traction separation law, TSL......................................................................................64
5.2 Models for brittle failure ................................................................................................65
5.2.1 Interface elements .......................................................................................................65
5.3 Constitutive model for timber under tension and shear .................................................66
5.4 Pure Tension Parallel to Grain .......................................................................................67
5.5 Applicabity of Model illustrated by Jörg Schmidt.........................................................69
5.6 Using ABAQUS for FEM..............................................................................................70
5.6.1 ABAQUS input for the cohesive element...................................................................70
Material parameters..............................................................................................................70
Mesh in the ABAQUS .........................................................................................................71
5.6.2 ABAQUS Analysis [Timber board without knot].......................................................72
5.6.2 ABAQUS Analysis [Timber board with knot]............................................................73
5.7 Element library...............................................................................................................75
5.8 Local coordinate system.................................................................................................76
5.9 Result..............................................................................................................................77
Chapter 6..................................................................................................................................81
Conclusion ...............................................................................................................................81
6.1 Limitation.......................................................................................................................81
6.2 Proposals for future work...............................................................................................81
REFERENCES.....................................................................................................................83

Chapter 1
Introduction
1.1 General remarks
Throughout the history of mankind, wood has been a material of vital importance. The field of
applications for wood and wood based products has been very wide. The first constructions in
which wood was used as a structural material were probably shelters. Before that, wood had
been used for weapons like clubs and spears, and as firewood. Later on, more than three
thousand years ago, the Egyptians produced veneers, laminates and paper from wood.
Accordingly, wood had and still has a very wide area of applications and is also one of the
oldest materials still in use, together with stone and bricks. Through the years, the use of wood
as a structural material has expanded greatly. There were still are many various types of
constructions where wood is the main raw material. Houses, bridges and vehicles for
transportation, such as boats, can be mentioned as examples. Wood has many advantages
compared to other materials. It is easily workable, and only small amounts of energy are
required to improve it. Also, it does not pollute as much as many other materials do.
Furthermore, it is a beautiful material that certainly should be used in buildings for more than
formwork timber.
Timber is dominant in many fields of application. The most important fields for sawn wood as
well as for wood based boards are building, carpentry, furniture and box making industries. Of
these the building industry has the greatest demand for wooden products. Although wood has
been used for thousands of years, there still is a great lack of knowledge about the behavior of
wood. The knowledge regarding making use of different species properties such as durability,
strength, toughness etc. is insufficient because of neglected research and education in wood
technology. To summarize, there are great needs regarding research in the timber industry, if
it is going to keep its share of the market. The products have to be improved, and better
knowledge about the material has to be acquired.
Wood is in general strong and stiff when loaded parallel to grain, but relatively weak when
loaded perpendicular to grain. The most troublesome modes of loading are commonly tension
perpendicular to grain and shear. Excessive loading in these modes causes perpendicular to
grain fracture and cracking along grain which may occur in a very brittle manner without much
prior warning by for example excessive deformations. Due to the strongly anisotropic strength
properties, an aim in design should be to avoid or at least limit loading in weak directions of
the material. Perpendicular to grain fracture is in general complicated to predict and there
appears to be a lack of knowledge regarding its modeling. This is in timber design codes of
practice reflected by absence of design criteria, or presence of questionable design criteria, for

structural elements exposed to perpendicular to grain tension and shear. Perpendicular to grain
fracture is a relatively common type of damage for timber structures.
Wood exhibits its highest strength in tension parallel to the grain. Tensile strength parallel to
the grain of small clear specimens is approximately 2 to 3 times greater than compressive strain
parallel to the grain, about 1.5 times greater than static bending strength and 10 to 12 times
greater than shear strength. Although tensile strength parallel to the grain is an important
property, it has not been fully determined for all commercial wood species due to several
reasons. First, having very low shear strength, wood has a tendency to break in shear or
cleavage at the fasteners and joints. Second, knots and growth defects have a great effect in
lowering the strength of wood subjected to tension parallel to the grain. In addition, the
manufacture of the test specimens is not easy and requires a lot of skilled manual labor.
Nevertheless, with the development of better mechanical fasteners and synthetic adhesives for
wood, higher proportion of the tensile strength can be utilized in modern design of wood
structures. Thus, tension parallel to grain properties of solid wood, as well as wood modified
by different treatments, should be further investigated.
1.2 Objectives
At the institute of Structural analysis, numerical methods and material models for the
simulation of timber structures by means of the Finite Element Method [FEM] were developed
in the past. By means of these models, the mechanical behavior and failure of the perfect timber
can be analyzed. The models were enhanced by methods to describe branches and knots in a
linear elastic FE analysis as well. In previous research, these approaches were combined with
existing material models to analysis failure initiated by knots. Brittle failure under tensile
loading is modelled using cohesive elements which has to be generated in between continuum
elements. Therefore, a new meshing method, so called Streamline Meshing, was developed.
This meshing procedure shall be applied to analysis timber boards under tensile loading to
simulate failure initiated by knots. In addition, FE meshes obtained in a previous project work
by means of a commercial FE software, namely SIMULIA ABAQUS, is used in comparison.
The cohesive elements have to be generated not only in between knots and surrounding wood
but also along possible crack paths. The cohesive material model has been developed to
represent tensile and shear failure perpendicular to grain since these failure modes are more
likely than longitudinal tensile failure in fiber direction. Regarding timber boards under tensile
loading this type of failure occurs as well.

The tasks for the project are
 Identify the basic failure mechanism for longitudinal tensile failure in perfect wood,
free of knots. By means of a literature study. Both micro and macroscopic effects are
interest, although focus on the later one.
 Checking of the applicability of the existing cohesive material model. If necessary, the
model has to be enhanced.
 Failure starting in knots occurs due to a combination of all failure modes. So, identify
the basic failure mechanism for timber boards containing knot exposed to tensile
loading.
 Therewith, a method to identify possible crack paths can be developed.
1.3 Organization of the Project
The project is organized in 6 chapters as follows
 Chapter 1: Introduction,
 Chapter 2: Properties of Timber,
 Chapter 3: Basic Fracture Mechanics and Modelling of Timber,
 Chapter 4: Effect of Knots in Timber,
 Chapter 5: Finite Element Modelling of Timber Boards with Knots,
 Chapter 6: Conclusion.
In chapter 1, we introduced timber with some basic properties. Then, the objectives and
organization of the project are outlined. Chapter 2 gives an overview of the typical structural,
physical, and mechanical timber properties. Chapter 3 provides an overview of up to date
models of timber. Chapter 4 represents the core of the project by focusing on knot effects. It
describes the strength changes of timber due to knots. Chapter 5 concerns use of SIMULIA
ABAQUS software and discussion of obtained results. Chapter 6 summarizes the important
ideas and results of this project together with suggestions for future work.

Chapter 2
Properties of Timber
Timber properties can be of physical, mechanical, chemical, biological or technological
essence. This chapter describes basic principles of timber behavior. It will discuss macroscopic
and microscopic structure of wood, mechanical properties and natural defects affecting
mechanical properties of wood.
“In the Figure 2. 1 it shows the chain extending from a micro to a macro level together with
the respective levels of modelling. At the micro level, such important factors as fiber shape,
cell wall thickness and micro fibril angle are considered. The properties of clear wood can be
described in terms of these factors, combined with growth characteristics. Starting with the
clear wood properties and taking such log imperfections as knots, spiral grain and the like into
account, one can define the behavior of sawn and dried timber rather precisely by use of proper
models” [Holmberg, 1998].
Figure 2.1: Modelling chain for wood extending from ultra-structure to end-user
products. [Holmberg, 1998]
2.1 Macrostructure and Microstructure of Wood
“Before analyzing the timber’s structural properties it is needed to understand wood anatomy
and structure. This can be considered at two levels: the microstructure, which can be examined
only with the aid of a microscope, and the macrostructure, which is normally visible to the
unaided eye” [Kuklík, 2008].
2.1.1 Macrostructure
The cross section of a tree can be divided into three basic parts: bark, cambium, and wood.
Bark  “The outer layer of a trunk. Protects the tree from fire, injury or temperature. The inner
layers of the bark transport nutrients from leaves to growth parts” [Kuklík, 2008].

Cambium  “Wood cells grow in a cambium. New wood cells grow towards the interior and
new bark cells grow towards the exterior of the cambium” [Kuklík, 2008].
Sapwood  “New cells of upward flow of sap [water and nutrients] from the roots to the crown
sapwood” [Kuklík, 2008].
Heartwood  “Cells in the inner part of the stem do not grow anymore and have the role of
receptacles of waste products [extractives]. Heartwood is darker in color than sapwood due to
the incrustation with organic extractives. Thank to these chemicals, heartwood is more resistant
to decay and wood boring insects. Heartwood formation results in reduction in moisture
content” [Kuklík, 2008].
Juvenile Wood  “The wood of the first 5 – 20 growth rings and thus it is a very early wood.
It has different physical and anatomical properties than that of mature wood. The differences
consist in fibril angle, cell length, and specific gravity, percentage of latewood, cell wall
thickness and lumen diameter. It tends to be inferior in density and cell structure and exhibits
much greater longitudinal shrinkage than mature wood” [Kuklík, 2008].
Pith  “The very center of the trunk. This part is typically of a dark color and represents the
original twig of a young tree” [Kuklík, 2008].
Figure 2.2: Cross section of a tree trunk. [Kuklík, 2008]

2.1.2 Microscopic structure of wood
In [Holmberg, 1998] it has shown a finer scale [that of microns] wood is a fiber-reinforced
composite. The cell walls are made up of fibers of crystalline cellulose embedded in a matrix
of amorphous hemicellulose and lignin, rather like the glass-fiber-in-polymer composite used
to make the hollow shaft of a fiber glass tennis racquet. The cells are hollow, tube-like
structures with the longitudinal axis approximately parallel to the tree stem axis. An example
of cell structure within one growth ring is found in Figure 2.3. The clear wood mechanical
properties are governed by the mechanical characteristics on the micro scale, i.e. by the cell
structure and the properties of the cell wall components.
Figure 2.3: Characteristics of a growth ring of spruce. [a] Cell structure. [b] Density
distribution. [Holmberg, 1998]
“The fundamental differences between woods are founded on the types, sizes, proportions, pits,
and arrangements of different wood cells. These fine structural details can affect the use of a
wood. This subchapter describes wood cells structure and function with regard to its differences
between hardwood and softwood” [Holmberg, 1998].
Wood cells
“Wood is composed of discrete cells connected and interconnected in an intricate and
predictable fashion to form an integrated continuous system from root to twig. The cells of
wood are usually many times longer than wide and are oriented in two separate systems: the
axial system [long axes running up and down the trunk] and the radial system [elongated
perpendicularly to the long axis of the organ and are oriented from the pith to the bark]. The
axial system provides the long-distance water movement and the bulk of the mechanical
strength of the tree. The radial system provides lateral transport for biochemical and an
important fraction of the storage function” [Wiedenhoeft, 2010].

“In most cells in wood there are two domains; the cell wall and the lumen. The lumen is a
critical component of many cells in the context of the amount of space available for water
conduction or in the context of a ratio between the width of the lumen and the thickness of the
cell wall. The lumen has no structure as it is the void space in the interior of the cell”
[Wiedenhoeft, 2010].
“Cell walls in wood give wood the majority of its properties. The cell wall itself is a highly
regular structure. The cell wall consists of three main regions: the middle lamella, the primary
wall, and the secondary wall [Figure 2.4]. In each region, the cell wall has three major
components: cellulose micro fibrils, hemicelluloses, and a matrix or encrusting material.
Generally, cellulose is a long string-like molecule with high tensile strength, micro fibrils are
collections of cellulose molecules into even longer, stronger thread-like macromolecules.
Lignin is a brittle matrix material. The hemicelluloses help link the lignin and cellulose into a
unified whole in each layer of the cell wall” [Wiedenhoeft, 2010].
Figure 2.4: Cut-away drawing of the cell wall: middle lamella [ML], primary wall with
random orientation of the cellulose micro fibrils [P], the secondary wall composed of its
three layers with illustration of their relative thickness and the micro fibril angle: [S1],
[S2], and [S3]; The lower portion of the illustration shows bordered pits in both
sectional and face view. [Wiedenhoeft, 2010]
“The primary wall [Figure 2.4] is characterized by a largely random orientation of cellulose
micro fibrils where the micro fibril angle ranges from 0° to 90° relative to the long axis of the
cell. The secondary cell wall is composed of three layers. The secondary cell wall layer S1 is a
thin layer and is characterized by a large micro fibril angle. The angle between the mean micro
fibril direction and the long axis of the cell is large [50° to 70°].The next wall layer S2 the most

important cell wall layer in determining the wood properties at a macroscopic level. This is the
thickest secondary cell wall layer characterized by a lower lignin percentage and a low micro
fibril angle [5° to 30°]. The S3 layer is relatively thin. The micro fibril angle of this layer is
>70°. This layer has the lowest percentage of lignin of any of the secondary wall layers”
“Communication and transport between the wood cells is provided by pits. Pits are thin areas
in the cell walls [cell wall modification] between two cells having three domains: the pit
membrane, the pit aperture and the pit chamber [Figure 2.4]” [Wiedenhoeft, 2010].
Softwood and hardwood
“Commercial timber is obtained from two categories of plants, hardwoods and softwoods. To
define them botanically, softwoods come from gymnosperms [mostly conifers] and hardwoods
come from angiosperms [flowering plants]. Softwoods are generally needle-leaved evergreen
trees such as pine and spruce, whereas hardwoods are typically broadleaf deciduous trees such
as maple, birch and oak. Main distinction between these two groups consists in their component
cells. Softwoods have a simpler basic structure that comprises only of two cell types with
relatively little variation in structure within these cell types. Hardwoods have greater structural
complexity consisting in greater number of basic cell types and a far greater variability within
the cell types. Hardwoods have characteristic type of cell called a vessel element [pore]
whereas softwoods lack these” [Wiedenhoeft, 2010].
Softwood
“Softwood structure is relatively simple [Figure 2.5]. The axial [vertical] system is composed
mostly of axial tracheids. The radial [horizontal] system is the rays, which are composed mostly
of ray parenchyma cells. Another cell types that can be present in softwood are axial
parenchyma and resin canal complex” [Wiedenhoeft, 2010].
Figure 2.5: Structure of softwood, magnified 250 times. [Wiedenhoeft, 2010]

“Tracheids are long cells being the major component of softwoods, making up over 90% of the
volume of the wood. They serve both the conductive and mechanical needs. Within a growth
ring, they are thin-walled in the early wood and thicker-walled in the latewood. Water flows
between tracheids by passing through circular bordered pits that are concentrated in the ends
of the cells. Pit membrane ensures resistance to flow. Tracheids are less efficient conduits
compared with the conducting cells of hardwoods due to the resistance of the pit membrane
and the narrow diameter of the lumina” [Wiedenhoeft, 2010].
“In evolving tracheids from earlywood to latewood, the cell wall becomes thicker, while the
cell diameter becomes smaller. The difference in growth may result in a ratio latewood density
to early wood density of 3:1” [Wiedenhoeft, 2010].
“Another cell type that is sometimes present in softwoods is axial parenchyma. Axial
parenchyma cells are vertically oriented and similar in size and shape to ray parenchyma cells.
Resin canal complex is present radially or axially in species of pine, spruce, Douglas-fir, and
larch. These structures are voids in the wood. Specialized parenchyma cells producing resin
surround resin canals. Rays are formed by ray parenchyma cells [brick-shaped cells]. They
function primarily in synthesis, storage, and lateral transport of biochemical and water”
“The cell arrangement in radial and tangential direction is different. Cells in radial direction
are assembled in straight rows while in tangential direction they are disordered [Figure 2.6].
This causes that tangential stiffness is lower than the radial one. Also, ray cells aligned in the
radial direction reinforce the structure radially and increase the radial stiffness” [Wiedenhoeft,
2010].
Figure 2.6: Cell structure arrangement in the radial and tangential directions.
[Wiedenhoeft, 2010]

Hardwood
“The structure of a typical hardwood [Figure 2.7] is more complicated than that of softwood.
The axial system is composed of various fibrous elements, vessel elements, and axial
parenchyma. As in softwoods, rays [composed of ray parenchyma] comprise the radial system,
but hardwoods show greater variety in cell sizes and shapes” [Kuklík, 2008].
Figure 2.7: Structure of hardwood, magnified 250 times. [Kuklík, 2008]
“Vessel elements [forming vessels] are the specialized water-conducting cells of hardwoods.
Vessels are much shorter than tracheids and can be arranged in various patterns. If all the
vessels are the same size and more or less scattered throughout the growth ring, the wood is
diffuse porous. If the earlywood vessels are much larger than the latewood vessels, the wood
is ring porous” [Wiedenhoeft, 2010].
“Hardwoods have perforated tracheary elements [vessels elements] for water conduction,
whereas softwoods have imperforate tracheary elements [tracheids]. Hardwood fibers have
thicker cell wallsand smaller lumina than softwood tracheids. Differences in wall thickness and
lumen diameters between earlywood and latewood are not as distinct as in softwoods” [Kuklík,
2008].
2.2 Physical Properties of Wood
“Physical properties describe the quantitative characteristics of wood and its behavior to
external influences other than applied forces. Included are such properties as moisture content,
density, dimensional stability, thermal and pyro lytic [fire] properties, natural durability, and
chemical resistance. Familiarity with physical properties is important because those properties
can significantly influence the performance and strength of wood used in structural
applications”[Ritter, 1990].

2.2.1 Directional Properties
“Wood is an orthotropic material because of the orientation of the wood fibers, and the manner
in which a tree increases in diameter as it grows, properties vary along three mutually
perpendicular axes: longitudinal [L], radial [R], and tangential [T]. The longitudinal axis is
parallel to the grain direction, the radial axis is perpendicular to the grain direction and normal
to the growth rings, and the tangential axis is perpendicular to the grain direction and tangent
to the growth rings [Figure 2.8]. Although wood properties differ in each of these three
directions, differences between the radial and tangential directions are normally minor
compared to their mutual differences with the longitudinal direction. As a result, most wood
properties for structural applications are given only for directions parallel to grain
[longitudinal] and perpendicular to grain [radial and tangential]” [Ritter, 1990].
Figure 2.8: The three principal axes of wood with respect to grain direction and growth
rings. [Ritter, 1990]
2.2.2 Moisture content
“The moisture content of wood [MC] is defined as the weight of water in wood given as a
percentage of oven dry weight:
MC =
[ ].
x 100
Where =moist weight, = dry weight and =oven dry weight
Depending on the species and type of wood, the moisture content of living wood ranges from
approximately 30 percent to more than 250 percent [two-and-a-half times the weight of the
solid wood material]. In most species, the moisture content of the sapwood is higher than that
of the heartwood” [Ritter, 1990].

2.3 Mechanical Properties of Timber
“Mechanical properties describe the characteristics of a material in response to externally
applied forces. They include elastic properties, which measure resistance to deformation and
distortion, and strength properties, which measure the ultimate resistance to applied loads.
Mechanical quantities are usually given in terms of stress [force per unit area] and strain
[deformation per unit length]” [Ritter, 1990].
“The basic mechanical properties of wood are obtained from laboratory tests of small, straight-
grained, clear wood samples free of natural growth characteristics that reduce strength.
Although not representative of the wood typically used for construction, properties of these
ideal samples are useful for two purposes. First, clear wood properties serve as a reference
point for comparing the relative properties of different species. Second, they may serve as the
source for deriving the allowable properties of visually graded sawn lumber used for design”
[Ritter, 1990].
2.3.1. Elastic Properties
“Elastic properties is material’s resistance to deformation under an applied stress to the ability
of the material to regain its original dimensions when the stress is removed. For an ideally
elastic material loaded below the proportional [elastic] limit, all deformation is recoverable,
and the body returns to its original shape when the stress is removed. Wood is not ideally
elastic, in that some deformation from loading is not immediately recovered when the load is
removed; however, residual deformations are generally recoverable over a period of time.
Although wood is technically considered a viscoelastic material, it is usually assumed to
behave as an elastic material for most engineering applications, except for time-related
deformations [creep]” [Ritter, 1990].
“For an isotropic material with equal properties in all directions, elastic properties are described
by three elastic constants: modulus of elasticity [E], shear modulus [G], and Poisson’s ratio
[µ]. Because wood is orthotropic, 9 constants are required to describe elastic behavior: 3moduli
of elasticity, 3 moduli of rigidity, and 3 Poisson’s ratios. These elastic constants vary within
and among species and with moisture content and specific gravity. The only constant that has
been extensively derived from test data, or is required in most bridge applications, is the
modulus of elasticity in the longitudinal direction. Other constants may be available from
limited test data but are most frequently developed from material relationships or by regression
equations that predict behavior as a function of density” [Ritter, 1990].
Modulus of Elasticity, E
“Modulus of elasticity relates the stress applied along one axis to the strain occurring on the
same axis. The three moduli of elasticity for wood are denoted EL, ER, and ET to reflect the
elastic moduli in the longitudinal, radial, and tangential directions, respectively. For example,

EL, which is typically denoted without the subscript L, relates the stress in the longitudinal
direction to the strain in the longitudinal direction” [Ritter, 1990].
Shear Modulus, G
“Shear modulus relates shear stress to shear strain. The three shear moduli for wood are denoted
GLR, GLT and GRT for the longitudinal-radial, longitudinal-tangential, and radial-tangential
planes, respectively. For example, LR is the shear modulus based on the shear strain in the LR
plane and the shear stress in the LT and RT planes” [Ritter, 1990].
Poisson’s Ratio, µ
“Poisson’s ratio relates the strain parallel to an applied stress to the accompanying strain
occurring laterally. For wood, the three Poisson’s ratios are denoted by μ ,μ andμ . The
first letter of the subscript refers to the direction of applied stress, the second letter the direction
of the accompanying lateral strain. For example, μ is Poisson’s ratio for stress along the
longitudinal axis and strain along the radial axis” [Ritter, 1990].
2.3.2 Strength Properties
“Strength properties describe the ultimate resistance of a material to applied loads. They
include material behavior related to compression, tension, shear, bending, torsion, and shock
resistance. As with other wood properties, strength properties vary in the three primary
directions, but differences between the tangential and radial directions are relatively minor and
are randomized when a tree is cut into lumber. As a result, mechanical properties are
collectively described only for directions parallel to grain and perpendicular to grain” [Ritter,
1990].
Compression
“Wood can be subjected to compression parallel to grain, perpendicular to grain, or at an angle
to grain [Figure 2.9]. When compression is applied parallel to grain, it produces stress that
deforms [shortens] the wood cells along their longitudinal axis. According to the straw analogy
each cell acts as an individual hollow column that receives lateral support from adjacent cells
and from its own internal structure. At failure, large deformations occur from the internal
crushing of the complex cellular structure. The average strength of green, clear wood
specimens of coast Douglas-fir and loblolly pine in compression parallel to grain is
approximately 26089 and 24207 kPa, respectively” [Ritter, 1990].

Figure 2.9: Compression in wood members. [Ritter, 1990]
“When compression is applied perpendicular to grain, it produces stress that deforms the wood
cells perpendicular to their length. Again recalling the straw analogy, wood cells collapse at
relatively low stress levels when loads are applied in this direction. However, once the hollow
cell cavities are collapsed, wood is quite strong in this mode because no void space exists.
Wood will actually deform to about half its initial thickness before complete cell collapse
occurs, resulting in a loss in utility long before failure” [Ritter, 1990].
“For compression perpendicular to grain, failure is based on the accepted performance limit of
1.016 mm deformation. Using this convention, the average strength of green, clear wood
specimens of coast. Douglas-fir and loblolly pine in compression perpendicular to grain is
approximately 4826 and 4557 kPa, respectively. Compression applied at an angle to grain
produces stress acting both parallel and perpendicular to the grain. The strength at an angle to
grain is therefore intermediate to these values and is determined by a compound strength
equation [the Hankinson formula]” [Ritter, 1990].
Tension
“The mechanical properties for wood loaded in tension parallel to grain and for wood loaded
in tension perpendicular to grain differ substantially [Figure 2.10]. Parallel to its grain, wood
is relatively strong in tension. Failure occurs by a complex combination of two modes, cell-to-
cell slippage and cell wall failure. Slippage occurs when two adjacent cells slide past one
another, while cell wall failure involves a rupture within the cell wall. In both modes, there is

little or no visible deformation prior to complete failure. The average strength of green, clear
wood specimens of interior-north Douglas-fir and loblolly pine in tension parallel to grain is
approximately 107558 and 79979 kPa, respectively” [Ritter, 1990].
Figure 2.10: Tension in wood members. [Ritter, 1990]
“In contrast to tension parallel to grain, wood is very weak in tension perpendicular to grain.
Stress in this direction acts perpendicular to the cell lengths and produces splitting or cleavage
along the grain that significantly affects structural integrity. Deformations are usually low prior
to failure because of the geometry and structure of the cell wall cross section. Strength in
tension perpendicular to grain for green, clear samples of coast Douglas-fir and loblolly pine
averages 2068 and 1792 kPa, respectively. However, because of the excessive variability
associated with tension perpendicular to grain, situations that induce stress in this direction
must be recognized and avoided in design” [Ritter, 1990].
Shear
"There are three types of shear that act on wood: vertical, horizontal, and rolling [Figure
2.11].Vertical shear is normally not considered because other failures, such as compression
perpendicular to grain, almost always occur before cell walls break in vertical shear. In most
cases, the most important shear in wood is horizontal shear, acting parallel to the grain. It
produces a tendency for the upper portion of the specimen to slide in relation to the lower
portion by breaking intercellular bonds and deforming the wood cell structure. Horizontal shear
strength for green, small clear samples of coast Douglas-fir and loblolly pine averages 6232
and 5950 kPa, respectively” [Ritter, 1990].

Figure 2.11: Shear in wood members. [Ritter, 1990]
Bending
“When wood specimens are loaded in bending, the portion of the wood on one side of the
neutral axis is stressed in tension parallel to grain, while the other side is stressed in
compression parallel to grain [Figure 2.12]. Bending also produces horizontal shear parallel to
grain, and compression perpendicular to grain at the supports. A common failure sequence in
simple bending is the formation of minute compression failures followed by the development
of macroscopic compression wrinkles. This effectively results in a sectional increase in the
compression zone and a section decrease in the tension zone, which is eventually followed by
tensile failure. The ultimate bending strength of green, clear wood specimens of coast Douglas-
fir and loblolly pine are reached at an average stress of 52848 and 50331 kPa, respectively”
[Ritter, 1990].
Figure 2.12: Bending in wood members produces tension and compression in the
extreme fibers, horizontal shear, and vertical deflection. [Ritter, 1990]

Torsion
“Torsion is normally not a factor in timber structure design, and little information is available
on the mechanical properties of wood in torsion. Where needed, the torsional shear strength of
solid wood is usually taken as the shear strength parallel to grain. Two-thirds of this value is
assumed as the torsional strength at the proportional limit” [Ritter, 1990].
Shock Resistance
“Shock resistance is the ability of a material to quickly absorb, then dissipate, energy by
deformation. Wood is remarkably resilient in this respect and is often a preferred material when
shock loading is a consideration. Several parameters are used to describe energy absorption,
depending on the eventual criteria of failure considered. Work to proportional limit, work to
maximum load, and toughness [work to total failure] describe the energy absorption of wood
materials at progressively more severe failure criteria” [Ritter, 1990].
2.4 Failure types
2.4.1 Compression
“Three basic failure patterns can be distinguished for compression perpendicular to grain
according to growth rings orientation and direction of load: crushing of earlywood, buckling
of growth rings and shear failure [Figure 2.13]” [Gibson and Ashby, 1988].
Figure 2.13: Failure types in compression perpendicular to the grain: [a]. crushing of
earlywood under radial loading, [b]. buckling of growth rings under tangential loading,
[c]. shear failure under loading at an angle to the growth rings. [Gibson and Ashby,
1988]
“Failure modes that occur during a compression test in longitudinal direction are crushing [the
plane of rupture is approximately horizontal], wedge split, shearing [the plane rupture makes
an angle of more than 45° with the top of the specimen], splitting [usually occurs in specimens
having internal defects prior to test], compression and shearing parallel to grain [usually occurs
in cross-grained pieces] and brooming or end-rolling [usually associated to an excessive MC
at the ends of the specimen or improper cutting of the specimen], see Figure 2.14. The failure
modes of splitting, compression and shearing parallel to grain and brooming or end-rolling are

the basis for excluding the specimen from the set of measured results” [Gibson and Ashby,
1988]
Figure 2.14: Failure types in compression parallel to grain: [a] crushing, [b] wedge split,
[c] shearing, [d]splitting, [e] Compression and shearing parallel to grain, [f] brooming
[end-rolling]. [Gibson and Ashby, 1988]
2.4.2 Tension
“Tensile loading perpendicular to the grain gives three failure patterns [similarly to
compression perpendicular to grain, [Figure 2.13]
 Tensile fracture in earlywood [radial loading].
 Failure in wood rays [tangential loading].
 Shear failure along growth ring [loading at an angle to the growth rings].
Crack propagation for opening mode [I] can occur in two ways: cell-wall breaking [crack
propagates across the cell wall] and cell-wall peeling [crack propagates between two adjacent
cells], see Figure 2.15” [Gibson and Ashby, 1988].
Figure 2.15: Crack propagation for opening mode [I] loading: cell-wall breaking [a],
cell-wall peeling [b]. [Gibson and Ashby, 1988]

“Failure in tension parallel to the grain follows one of the patterns shown in Figure 2.16,
namely shear, a combination of shear and tension, pure tension and splinter mode. After the
destructive tests, and confirming the theoretical results expected, the patterns observed on
Figure 2.17 and Figure 2.18 were observed” [Gibson and Ashby, 1988].
Figure 2.16: Theoretically possible failure patterns: [a] splinter, [b]shear and tension
failure, [c] shear failure; and [d] pure tension failure. [Gibson and Ashby, 1988]
The parallel to grain tensile strength is the conventional value determined by the maximum
strength applied to a specimen. Each load-extension curve was reduced to a true stress-true
strain plot; from these, yield strengths were determined using a strain displacement that was
equivalent to a 0.3% offset in the usual terminology.
Figure 2.17: Typical failure patterns observed: [a] splinter, and [b] shear and tension
failure. [Gibson and Ashby, 1988]

Figure 2.18: Other typical failure patterns observed in tensile tests: [a]shear failure, and
[b] pure tension failure. [Gibson and Ashby, 1988]
2.4.3 Stress-strain curves
“Typical stress-strain curves for dry wood loaded in longitudinal [L], radial [R] and tangential
[T] direction in compression and in tension in L direction are presented in Figure 2.19”
[Holmberg, 1998].
Figure 2.19: Typical stress-strain curves for wood loaded in compression in L, R and T
direction and for tension in L direction. [Holmberg, 1998]
“Development of the stress-strain curves in L, T and R [longitudinal, transversal and radial]
compression show an initial elastic region, followed by a plateau region and a final region of
rapidly increasing stress. The yield stresses for T and R compression are about equal and are
considerably lower than L compression. R compression is characterized by a small drop in
stress after the end of elastic region and it has slightly irregular plateau compared to the smooth
plateau of T compression and serrated plateau region of L compression” [Holmberg, 1998].

2.4.4 Fracture
“In fracture mechanics, three general fracture modes are defined: symmetric opening
perpendicular to the crack surface [I], forward shear mode [II], and transverse shear mode [III],
see Figure 2.20.Modes [II] and [III] involve anti symmetric shear separations” [Kretschmann,
2010].
Figure 2.20: Failure modes in wood: opening mode [I], forward shear mode [II] and
transverse shear mode [III]. [Kretschmann, 2010]
“In wood, eight crack-propagation systems can be distinguished: RL, TL, LR+, LR-, TR+, TR-
, LT,and RT. The first letter of the crack-propagation system denotes perpendicular direction
to the crack plane and the second one refers to direction in which the crack propagates. The
distinction between + and – direction arises because of the asymmetric structure of the growth
rings, see Figure 2.21. For each of eight crack-propagation systems, fracture can occur in three
modes and thus cracks in wood can arise in 24 different principal manners” [Gibson and
Ashby, 1988].
Figure 2.21: Eight modes of possible crack propagation in wood [Gibson and Ashby,
1988].
“It is suggested that fracture toughness is either insensitive to moisture content or increases as
the material dries [until maximum at MC of 6% - 15%]. Fracture toughness then decreases with
further drying” [Kretschmann, 2010].

Mode I fracture characteristics of one softwood [spruce] and three hardwoods [alder, oak and
ash in the crack propagation systems RL and TL are presented in Reiterer [2002]. Wedge
splitting test under loading perpendicular to grain was used [Figure 2.22]. Testing arrangement
is shown in Figure 2.23.
Figure 2.22: Wedge splitting test: specimen geometry and grain orientation [RL, TL].
[Reiterer, 2002]
Figure 2.23: Wedge splitting test: testing arrangement. [Reiterer, 2002]
“The load-displacement curves for different crack propagation systems are presented in Figure
2.24.Spruce shows stable crack propagation until complete separation of the specimens.
Hardwoods behave in a different manner: after macro-crack initiation at the maximum
horizontal splitting force sudden drop in the load–displacement curve occurs indicating
unstable crack propagation. This drop is followed by crack arresting leading to another

maximum. This is explained by the more brittle behavior of the hardwoods, which can be
attributed to the fact that hardwood fibers are shorter than spruce fibers and energy dissipating
processes [e.g. fiber bridging] are less effective. Also, less micro-cracks is formed during the
crack initiation phase for the hardwoods which can be shown by means of acoustic emission
measurements” [Reiterer, 2002].
Figure 2.24: Typical load-displacement curves obtained by the wedge splitting test in
the RL [a] and TL [b] systems. [Reiterer, 2002]
2.4.5 Fracture toughness and fracture energy
“The fracture mechanics approach has three important variables: applied stress, flaw size, and
fracture toughness while traditional approach to structural design has two main variables:
applied stress and yield or tensile strength. In the latter case, a material is assumed to be
adequate if its strength is greater than the expected applied stress. The additional structural
variable in fracture mechanics approach is flaw size and fracture toughness. They replace
strength as the relevant material property. Fracture mechanics quantifies the critical
combinations of the three variables” [Anderson, 2005].
“In fracture mechanics, fracture toughness is essentially a measure of the extent of plastic
deformation associated with crack extension. Fracture toughness is measured by critical strain
energy release rate according to energy-balance approach or by critical stress intensity
factor [SIF] according to stress intensity approach [Dinwoodie, 1981]. In case linear elastic
fracture mechanics [LEFM] is involved, critical strain energy release rate Gc is equal to fracture
energy [ = ]. Both variables are a material property that gives information about
when a crack starts propagating [Bostrom,1992]. These subchapters describe material
properties and a few examples of current test methods available for their determination”
[Anderson, 2005].
Critical strain energy release rate [energy-balance approach]
“The energy approach assumes that crack extension [i.e. fracture] occurs when the energy
available for crack growth is sufficient to overcome the resistance of the material. The material
resistance may include the surface energy, plastic work, or other types of energy dissipation

associated with crack propagation. This approach is based on energy release rate G which is
defined as the rate of change in potential energy with the crack area for a linear elastic material.
At the moment of fracture, energy release rate is equal to its critical value [ = ] which is
a measure of fracture toughness” [Anderson, 2005].
“For a crack of length 2 in an infinite plate [where width of the plate is >>2 ] subjected to a
remote tensile stress [Figure 2.25], the energy release rate is expressed by
= [2.1]
Where is modulus of elasticity is, is remotely applied stress, and is the half-crack length.
If fracture occurs [ = ], the Eq. [2.2] describes the critical combinations of stress and crack
size for failure:
= [2.2]
The energy release rate G is the driving force for fracture while Gc is the material’s resistance
to fracture. Fracture toughness is independent of the size and geometry of the cracked body
and thus a fracture toughness measurement on a laboratory specimen should be applicable to
structure. These assumptions are valid as long as the material behavior is predominantly linear
elastic” [Anderson, 2005].
Figure 2.25: Through-thickness crack in an infinite plate [plate width is >>2a] subject to
a remote tensile stress. [Anderson, 2005]

Critical stress intensity factor [SIF]. [Stress intensity approach]
“Stress intensity approach examines the stress state near the tip of a sharp crack and defines
critical stress intensity factor that is a fracture toughness measure and it can be used for
normal opening crack modes I and shear sliding modes II and III [ , , ]. The text
of this subchapter describes equations only for opening crack failure mode I. Figure 2.26
schematically shows an element near the tip of a crack in an elastic material, together with the
in-plane stresses on this element. Each stress component is proportional to stress intensity
factor for fracture mode I. If material fails locally at some critical combination of stress and
strain, then fracture must occur at a critical stress intensity factor ” [Anderson, 2005].
Figure 2.26: Stresses near the tip of a crack in an elastic material. [Anderson, 2005]
“For an infinite plate [Figure 2.25], the stress intensity factor is given by
= √ [2.3].
Failure occurs when = where KI is the driving force for fracture and is a measure
of material resistance. KIC is assumed to be a size-independent material property. If we compare
Eq. [2.1] and Eq. [2.3], we can derive relation between and
= [2.4].
This same relationship holds for GC and KIC. Thus, the energy and stress-intensity approaches
to fracture mechanics are essentially equivalent for linear elastic materials” [Anderson, 2005].
2.4.6 Fracture energy,
“Fracture energy [N/m] is an amount of energy required to form a unit area of a new crack
in the material. For opening crack mode I, , can be defined as the area under the stress-
displacement curve − for the fracture process zone as follow

, = ∫
,
[2.5]
where . is critical crack opening of the crack in normal direction to the crack [mm, is
actual crack opening of the crack in normal direction to the crack [mm] and is the stress
acting in normal direction at the crack. Similarly, fracture energy for pure shear mode II,
, can be defined as the area under the stress displacement curve − for the fracture
process zone as follows
, = ∫
,
[2.6]
where is critical crack opening of the crack in tangential direction to the crack [mm], is
actual crack opening of the crack in tangential direction to the crack [mm] and is the stress
acting in tangential direction to the crack”[Bostrom, 1992].
2.5 Strength, toughness, failure and fracture morphology
“There are two fundamentally different approaches to the concept of strength and failure. The
first is the classical strength of materials approach, attempting to understand strength and
failure of timber in terms of the strength and arrangement of the molecules, the fibrils, and the
cells by thinking in terms of a theoretical strength and attempting to identify the reasons why
the theory is never satisfied” [Peter, 2010].
“The second and more recent approach is much more practical in concept since it considers
timber in its current state, ignoring its theoretical strength and its microstructure and stating
that its performance will be determined solely by the presence of some defect, however small,
that will initiate on stressing a small crack; the ultimate strength of the material will depend on
the propagation of this crack. Many of the theories have required considerable modification for
their application to the different fracture modes in an anisotropic material such as timber. Both
approaches are discussed below for the more important modes of stressing” [Peter, 2010].
From the research [Holmberg, 1998] it can be seen as Figure 2.27 differences in types of
failure in the cell structures. On the basis of these we can design the micro macro modelling of
the cell structure for determining failure pattern.

Figure 2.27: Cell structure deformations at failure under various loading conditions. [a]
Compression, [b] tension, [c]shear; and [d]combined shear and compression.
[Holmberg, 1998]
2.5.1 Classical approach
Tensile strength parallel to the grain
“Over the years a number of models have been employed in an attempt to quantify the
theoretical tensile strength of timber. In these models it is assumed that the lignin and
hemicelluloses make no contribution to the strength of the timber; in the light of recent
investigations, however, this may not be valid for some of the hemicelluloses. One of the
earliest attempts modelled timber as comprising a series of endless chain molecules, and
strengths of the order of 8000 MPa were obtained” [Peter, 2010].
“More recent modelling has taken into account the finite length of the cellulose molecules and
the presence of amorphous regions. Calculations have shown that the stress needed to cause
chain slippage is generally considerably greater than that needed to cause chain scission,
irrespective of whether the latter is calculated on the basis of potential energy function or bond
energies between the links in the chain; preferential breakage of the cellulose chain is thought
to occur at the C–O–C linkage. These important findings have led to the derivation of minimum
tensile stresses of the order of 1000–7000 MPa” [Mark, 1967].
“As illustrated in Figure 2. 28, the degree of interlocking is considerably greater in the
latewood than in the earlywood. Whereas in the former, the fracture plane is essentially vertical,

in the latter the fracture plane follows a series of shallow zigzags in a general transverse plane;
it is now thought that these thin walled cells contribute very little to the tensile strength of
timber. Thus, failure in the stronger latewood region is by shear, while in the earlywood, though
there is some evidence of shear failure, most of the rupture appears to be transwall or brittle”
[Peter, 2010].
Figure 2.28: Tensile failure in spruce [Picae abies] showing mainly transverse cross-wall
failure of the earlywood [left] and longitudinal intra-wall shear failure of the latewood
cells [right] [magnification× 200, polarized light].[Peter, 2010]
Toughness
“Timber is a tough material, and in possessing moderate to high stiffness and strength in
addition to its toughness, it is favored with a unique combination of mechanical properties
emulated only by bone which, like timber, is a natural composite. Toughness is generally
defined as the resistance of a material to the propagation of cracks. In the comparison of
materials it is usual to express toughness in terms of work of fracture, which is a measure of
the energy necessary to propagate a crack, thereby producing new surfaces” [Peter, 2010].
“One of the earlier theories to account for the high toughness of timber was based on the work
of Cook and Gordon [1964], who demonstrated that toughness in fire reinforced materials is
associated with the arrest of cracks made possible by the presence of numerous weak interfaces.
As these interfaces open, so secondary cracks are initiated at right angles to the primary,

thereby dissipating the energy of the original crack. This theory is applicable to timber, as
Figure 2.29 illustrates, but it is doubtful whether the total discrepancy in energy between
experiment and theory can be explained in this way” [Peter, 2010].
Figure 2.29: Crack-stopping in a fractured rotor blade. The orientation of the
secondary cracks corresponds to the micro fibrillary orientation of the middle layer of
the secondary cell wall [magnification × 500, polarized light]. [Peter, 2010]
Fatigue
“Fatigue is usually defied as the progressive damage and failure that occur when a material is
subjected to repeated loads of a magnitude smaller than the static load to failure; it is, perhaps,
the repetition of the loads that is the significant and distinguishing feature of fatigue.
In fatigue testing the load is generally applied in the form of a sinusoidal or a square wave.
Minimum and maximum stress levels are usually held constant throughout the test, though
other wave forms, and block or variable stress levels, may be applied. The three most important
criteria in determining the character of the wave form are.
• The stress range,, where  = max -min
• The R-ratio, where R = min/max, which is the position of minimum stress [min] and
maximum stress [max] relative to zero stress. This will determine whether or not reversed
loading will occur. It is quantified in terms of the R ratio, e.g. a wave form lying symmetrically
about zero load will result in reversed loading and have an R ratio of -1 [The frequency of
loading]. The usual method of presenting fatigue data is by way of the S–N curve, where log
N [the number of cycles to failure] is plotted against the mean stress, S; a linear regression is
usually fitted.

Fracture mechanics has been applied to various aspects of timber behavior and failure, e.g. the
effect of knots, splits and joints, and good agreement has been found between predicted values
using fracture mechanics and actual strength values” [Peter, 2010].
At the conclusion from this chapter, we can have a look on the properties of timber which are
related to the strength of the timber and how does failure occur to the timber structures when
loads are applied. In the next chapter we will have a look on basic failure mechanics and
modelling of timber. From different publications mentioned in this chapter have more precise
details of these properties of the timber. But for this project we have included only the certain
things which are related to the topic of the project.

Chapter 3
Basic Fracture Mechanics and Modelling Of Timber
Numerical models for wood fracture and failure are commonly based on the finite element
method. Most of these models originate from general theoretical considerations for other
materials. This limits their usefulness because no amount of complexity in a model can
substitute for lack of inappropriate representation of the physical mechanisms involved. As for
other materials, wood fracture and failure models always require some degree of experimental
calibration, which can introduce ambiguity into numerical predictions because at present there
is a high degree of inconsistency in test methods. In this chapter we will try to look the types
of fracture mechanics used for timber and modelling of timber for analysis.
3.1 Introduction of the Fracture Mechanics
“Many materials, including wood, have preexisting flaws or discontinuities that grow when
subjected to certain stress conditions. Fracture mechanics relates the material properties, flaw
geometry, and applied loads to the resulting stress conditions surrounding the crack tip.
Fracture mechanics assumes cracks propagate by three basic fracture modes. In wood fracture,
Mode I [opening mode] and Mode II [forward shear] are most common. Mode III fracture
occurs in wood beams with side checks, but is more common in fiber-based materials such as
paper. In lumber, Mode I and Mode II fractures often occur together [mixed mode fracture]”
[Mallory, 1987].
“Crack propagation depends on the degree to which stress levels decay at distances away from
the crack tip. The stress-intensity factor, KI in Mode I or KII in Mode II fractures, is a parameter
that directly indicates the level of stress decay in the material surrounding the crack tip for a
given loading condition. The stress intensity factor associated with impending fracture in a
single fracture mode is defined as the critical stress intensity, KIC or KIIC. The critical stress-
intensity factor corresponds to a mode and direction of crack propagation and is considered a
basic property of the material. For wood this means the critical stress-intensity factors are a
function of species and affected by many of the same factors that affect other wood material
properties [e.g. specific gravity [SG]. and moisture content [MC]” [Mallory, 1987].
“Fracture is assumed to occur when the stress intensity factors are of sufficient magnitude to
satisfy a fracture criterion. Fracture criteria relate the mathematics associated with computation
of the stress intensity factors and material properties to real material fracture. In simple
problems involving only Mode I fracture, the criterion for fracture will be
=⁄ [3.1].

If the stress-intensity factor, KI, divided by the critical stress intensity, KIC, is less than one, the
stresses will redistribute and arrest the crack. If KI divided by KIC is greater than or equal to
unity, the crack will propagate. For a pure Mode II fracture, a similar criterion is provided by
substituting KII and KIIC in Equation [3.1]. In many practical problems involving wood
members, both Mode I and Mode II fracture occur together. In these mixed mode situations,
both Mode I and Mode II stress-intensity factors must be computed and assessed with a fracture
criterion that is a function of the Mode I and Mode II critical stress-intensity factors. Though a
particular mixed mode fracture criterion has not been thoroughly substantiated for wood, the
theory proposed by [Wu, 1967]. Wu’s criterion, which was developed on the basis of tests with
balsa wood, is of the following form” [Mallory, 1987].
⁄ + [ ⁄ ] = 1 [3.2]
3.2 Fracture mechanics models
The objective this part is to give a brief presentation of different fracture mechanics models
used by wood scientists and researchers today. Consequently no complete definitions or
derivations of the formulas and equations are given.
Usually when fracture mechanics is applied to wood, the linear elastic approach is employed.
In fact, the model gives relatively good results in many situations where large structures with
cracks are analyzed. However, there are situations where other models have to be applied. In
order to give an insight into some fracture mechanics models suitable for wood, three different
models are described. These models today are more or less applied to wood by different
researchers.
3.2.1 Linear elastic fracture mechanics models
“Linear elastic fracture mechanics [LEFM] models are continuum representations and usually
implemented by FEA. The concepts are only applicable for estimation of the load level that
will propagate an initially sharp crack. Thus, the concepts are unsuitable for predicting
development of cracking, especially for materials that develop toughening mechanisms once
cracks begin to grow. This can be quite problematic because wood and wood based materials
often embody heterogeneity that affects crack extension and promotes toughening” [Vasic,
Smith and Landis, 2004].
“For homogeneous orthotropic material with a crack lying on one plane of symmetry the stress
intensity factors [K values] are evaluated according to [Sihet, 1965] and applied within the
equation for crack growth K= Kc where Kc is the appropriate fracture toughness. Kc values are
considered to be material constants that can be obtained from the experiments with the
relationship = ∗ where Gc is the critical energy release rate and E* is the harmonic
elastic modulus. Orthotropic stress intensity factors, unlike their isotropic cousins, depend on

the elastic constants [Bowie and Freese 1972]. When a material is not a homogeneous
continuum at cellular or finer scales, it should be treated as heterogeneous [Kanninen et al.
1977]. There is, therefore, a strong element of educated judgment in any decision to apply
LEFM to wood” [Vasic, Smith and Landis, 2004].
A standard finite element program with quadratic isoperimetric elements can be modified to
extract stress-intensity factors with a rather simple scheme [Boone, Wawrzynek, 1987]. This
involves:
 Modifying the element stiffness matrix to include orthotropic stiffness constants
 Placing quarter-point elements at the crack tip
 Extracting displacements from the quarter-point elements at the crack faces
 Including a simple algorithm to interpret stress-intensity factors from the displacements
Figure 3.1: Barsoum’s 3D singular finite element. [Vasic, Smith and Landis, 2004]
Accurate computation is attained when the elements are regularly shaped and well distributed
around the crack tip. Barsoum’s elements and this type of procedure have been applied in a
number a wood fracture problems. As is known from general mechanics considerations,
provided geometric proportioning is held constant, the ratio of strain energy stored in a member
subjected to external load relative to the energy required for crack extension increases with any
increase in the member volume. This means that there is minimal load release when cracks
start to propagate and the possibility of crack stabilization is minimal even in the presence of
coarse inhomogeneity. Toughening around the crack tip has little influence for large systems
and members. [Vasic, Smith and Landis, 2004]
3.2.2 Non-linear elastic fracture models
“Non-linear elastic fracture mechanics [NLFM] methods need to be part of an analyst’s arsenal.
NLFM methods are sophisticated numerical prediction tools that have as their main advantage
the ability to predict post-peak stress fracture behavior”[Vasic, Smith and Landis, 2004].

Fictitious crack model, FCM
“The FCM is assumed advantageous over LEFM because no pre-existing crack is required and
it recognizes modes of energy dissipation other than creation of fracture surface. The concept
is that fracturing in a material introduces discontinuities in the displacement field. It is assumed
that damage is confined to a fracture plane of zero thickness. FEM implementation links or
continuous contact elements are used to connect nodes on opposite faces of existing or potential
crack planes [Figure 3.2]. Linking elements simulate experimental stress vs. crack width
relationships [r–w curves] such as that shown in Figure 3.2. Hence, the model is fictitious.
Many past studies have accepted that the FCM would fit the damage processes in wood despite
any explicit proof” [Vasic, Smith and Landis, 2004].
Figure 3.2: Fictitious crack model [FCM], is tensile strength, and is a crack
opening. [Vasic, Smith and Landis, 2004]
“The numerical results are usually presented as a load–displacement curve for a specimen or
structural component. It is assumed that once the crack opening is sufficient, spring stiffness
drops to zero and no stress transferring ability exists and a real as opposed to fictitious crack is
established. The FCM can be applied under combined stress conditions as has been illustrated
in the context of adhesive joints that produce softening in wood due to both tension
perpendicular to grain and shear parallel to grain analysis” [Wernersson 1990].
Bridged crack model
“Based on real-time observation of opening mode fracture processes in softwoods [Vasic
2000], it has been concluded that a bridged crack model [BCM] is a correct theoretical NLFM
representation of wood.

Figure 3.3: Application of the FCM to predict load–crack opening displacements.
[Vasic, Smith and Landis, 2004]
“Figure 3.4 gives a schematic of how the model is implemented. The conceptual difference
between FCM and BCM models concerns whether a stress singularity is permitted at the crack
tip. The BCM assumes that a stress singularity at a sharp crack tip co-exists with a bridging
zone behind the crack tip, i.e. the bridging zone is not fictitious as in the FCM. The main
assumptions of the BCM is that fracture occurs when the critical fracture toughness is reached
at the tip of the crack. The criterion for crack extension and opening is therefore the same as
for LEFM crack extension” [Vasic, Smith and Landis, 2004].
Figure 3.4: Bridged crack model (BCM). [Vatic, Smith and Landis, 2004]
“Thus, the fracture criterion is stress based and fracture toughening during crack growth can
be represented by simply adding the stress contributed from bridging fibers [or other
toughening mechanisms] to the net crack tip stress intensity” [Vasic, Smith and Landis,
2004].

Lattice fracture model
“This section discusses lattice models as an alternative to the more usual continuum-based
representations that are discussed above. Discrete elements within lattice arrangements
simulate real ultrastructure features. Therefore, it is straightforward to explicitly incorporate
heterogeneity and variability making lattice models a natural choice for representing disordered
materials [Curtin and Scher, 1990; Herrmann and Roux, 1990]. It follows that such models
can be used to represent wood that embodies both structured and random heterogeneity at
various length scales. Being morphology-based the modelling eliminates errors associated with
homogenization which occurs in continuum-based FEA. In the past lattice models have been
used mainly with concrete-based materials and incorporated both random and uniform lattice
geometry with uniform and variable elements” [Jirasek and Bazant, 1995; Schlangen, 1995;
Schlangen and Garboczi 1996, 1997].
“The material is represented as an array of nodes connected by a network of discrete beam or
spring elements. Figure 3.5 shows one possible discretization appropriate for wood.
Figure 3.5: Fracture toughness vs. crack length for an end-tapered DCB specimen.
[Vasic, Smith and Landis, 2004]
The longitudinal wood cells are represented by beam elements [large horizontal elements in
the Figure 3.6, while a network of diagonal spring elements simulates their connectivity. The
chosen size of a lattice cell in the specific example corresponds to a bundle of cells so that the
modelling is at the scale of wood growth rings. In genera l, models may be 2D or 3D and

Figure 3.6: Lattice finite element mesh for wood. [Vasic, Smith and Landis, 2004]
Elements defined on any appropriate scale. In order to account for pre-existing heterogeneities,
disorder of wood ultra-structure is introduced via statistical variation of element stiffness and
strength characteristics. Stiffness and strength characteristics can be assumed to fit a Gaussian
[or another] distribution with specified mean and standard deviation” [Vasic, Smith and
Landis, 2004].
“Lattice element properties are not chosen arbitrarily. As elaborated by Davids et al. [2003],
element properties are determined from matching the global lattice response to the orthotropic
elastic properties of wood in bulk” [Vasic, Smith and Landis, 2004].
Figure 3.7: Lattice fracture model vs. experimental tension perpendicular to grain
response under displacement control. [Vasic, Smith and Landis, 2004]

Figure 3.8: Typical lattice damage pattern tension perpendicular to grain response
under displacement control. [Vasic, Smith and Landis, 2004].
“The parameters of the model that can be adjusted are element aspect ratio, the angle that
defines the orientation of the diagonal members, and the mean stiffness of each type of element.
Optimal mean values of elastic constants are assumed to be those that minimize the normal
sized least-squares objective function of the orthotropic bulk wood values. Other properties
such as mean strengths and coefficients of variation are determined from an adjustment
procedure that matches experimental and nominal numerical bulk wood response in shear
parallel to grain, radial tension perpendicular to grain and tension parallel to grain. The research
effort in developing this numerical framework of LFM is still in progress and numerous issues
are yet to be resolved before the approach can achieve its full potential. Like all other fracture
models applied to wood the LFM does not yet recognize that wood embodies both structured
and unstructured in homogeneity” [Vasic, Smith and Landis, 2004].
3.3 Modelling of timber properties
Micro-macro modeling of wood properties
Two types of models of timber that can be used for an analysis of timber in the refining process
in mechanical pulp manufacture have been developed by [Holmberg, 1998].This application
Isa good illustration of modeling spanning from micro to macro scale. It involves large
deformations, plasticity, damage and fracture. Micro models of the cellular microstructure
[micro level] are used for analysis of individual fibers deformation. They are very general with
a very high degree of resolution, but they allow studying only very small pieces of wood. They
are also difficult to handle with the computer resources available today. Compared to micro
modeling, macro modeling [continuum modeling] is based on the average material properties
that can be obtained from a micro model. It allows analysis of deformation and fracturing of
large wood pieces. On the other hand, macro modeling does not permit analysis of the
deformation and fracturing of the individual fibers. The micro-macro model is based on an
experimental study of the defibration process [Figure 3.9] described in [Holmberg, 1998].The
behavior of a specimen is characterized by development of cracks and by large volumetric
changes in earlywood under compression.

Figure 3.9: Failure process in a 5 mm high wood specimen loaded perpendicularly to
grain by steel grips [simulation of refiner discs during pulp production]; Load-
displacement [horizontal, vertical] curve [Holmberg, 1998].
3.3.1 Micro-mechanical approach
“For the micro model of wood, equivalent stiffness and shrinkage were determined by a
homogenization method. The basic equations are solved by means of finite element method
[FEM]. The equivalent properties were determined in steps presented in Figure 3.10.
Figure 3.10: Modeling scheme of micro-mechanical approach. [Holmberg, 1998]

In the first step, equivalent properties of cell wall layers were calculated from the properties of
cellulose, hemicellulose and lignin. Micro fibril models were created for representing the
different layers of the cell wall. FEM together with homogenization approach were used to
determine the equivalent properties from these macro fibril models. Material stiffness’s were
transformed in order to relate the local directions of micro fibrils with the global L, R, and T
directions” [Holmberg, 1998].
“The aim of the second step was to determine the equivalent properties of wood structure. In
this step, the cell structure was modeled by means of a five-parameter cell structure model with
the most representative properties. For this purpose, 3D cell structures of complete growth
rings composed of irregular hexagonal cells were created [Figure 3.11a]. A model of a
complete growth ring was obtained with respect to the density function and the radial widths
of the cells [Figure 3.11b]. Density and cell wall thickness were assumed to increase slightly
linearly for earlywood, rapidly [quadratic ally] for transition zone and linearly in latewood
zone. Cell width in radial direction was considered constant for the earlywood, decreasing for
the transition wood and constant in the zone of latewood” [Holmberg, 1998].
Figure 3.11: Modeling of a growth ring: single-cell geometry [a], photographed and
modeled cell structures [b]. [Holmberg, 1998]
3.3.2 Continuum modeling approach
“To analyze mechanical behavior of wood on structural element scale [macro modeling], it is
desirable to model it as an equivalent continuum. However, it is necessary to take into account
various damage phenomena, such as defibration [fracture propagating along wood fibers]. In
order to perform a proper model of initial defibration by means of a continuum model,
[Holmberg, 1998] considered the following characteristics of wood:
 Variation in material properties within a growth ring,

 Nonlinear inelastic response of earlywood subjected to compression perpendicular to
the grain,
 Fracture behavior of material.
A Coulomb friction model was used for the interface elements between the wood specimen
and steel grips. The steel grips were modeled as rigid surfaces. A typical FE mesh that was
used is shown in Figure 3.12.
Figure 3.12: A typical finite element mesh used in the simulations [Holmberg, 1998].
Two specimen types were described: the wood subjected to shear loading in radial and in
tangential direction both in dry and wet conditions [Figure 3.13]. The deformation and fracture
process agree well with the experimental results” [Holmberg, 1998].
Figure 3.13: Comparison between numerical simulation and experimental results:
loading in tangential [a] and radial [b] direction. [Holmberg, 1998]

3.4 FEM at large deformations and brittle failure prediction
A constitutive model of wood based on both hardening associated with material densification
at large compressive deformations and brittle failure modes was developed by [Oudjene and
Khelifa, 2009]. Coupling between the anisotropic plasticity and the ductile densification was
considered.
“The model was implemented in the commercial software ABAQUS and its validation was
performed by means of uniaxial compressive test in longitudinal and radial direction and three
points bending [TPB] test. Material parameters [elasticity, plasticity hardening, densification]
were determined using experimental data [stress-strain curves] obtained from uniaxial
compression tests in longitudinal and radial direction. Distinction between radial and tangential
planes was disregarded. In two-dimensional finite element model was assumed isotropic
behavior in the transverse direction [radial and tangential].
The coupled model is well suited for analysis with large compressive deformations
perpendicular to the grain. The behavior is accurately predicted until 25% of deformation by
both the coupled and the uncoupled cases. The densification effect occurs beyond this limit and
is well predicted by the coupled model while the uncoupled one provides fairly good agreement
with the experiment.
The coupled, uncoupled and linear elastic models give almost the same results in linear load
displacement curves as the experiment in bending until a final failure. Hence, the effect of the
densification should be neglected since the plastic behavior is not significant. Linear elastic
material model is more accurate for the behavior after reaching the compressive yield stress in
perpendicular direction than coupled or uncoupled models.
The results obtained from the uniaxial compressive test demonstrate the capability of the model
to simulate the wood behavior at large compressive deformations and show clearly the effect
of the densification on the plastic behavior. The results obtained from the three-points bending
test show a good implementation of the brittle failure criterion and demonstrates the suitability
of the developed model to analyze and design wooden structures”[Oudjene and Khelifa,
2009].

Chapter 4
Effect of Knots in Timber
Knots reduce the strength of wood because they interrupt the continuity and direction of wood
fibers. They can also cause localized stress concentrations where grain patterns are abruptly
altered. The influence of a knot depends on its size, location, shape, soundness, and the type of
stress considered. In general, knots have a greater effect in tension than in compression,
whether stresses are applied axially or as a result of bending. Inter grown knots resist some
kinds of stress but encased knots or knotholes resist little or no stress. At the same time, grain
distortion is greater around an intergrown knot than around an encased knot of equivalent size.
As a result, the overall effects of each are approximately the same.
4.1 Knots
“Knots are remnants of branches in the tree appearing in sawn timber. Independent of the cut
of the board, knots occur in two basic varieties: intergrown knots and encased knots. If the
branch was alive at the time when the growth rings making up a board were formed, the wood
of the trunk and that branch is continuous; this is referred to as intergrown knot [Figure 4.1a].
If the branch was dead at the time when growth rings of a board were formed, knot is not
continuous with the stem wood; this produces an encased knot [Figure 4.1b]. Encased knots
generally disturb the grain angle less than intergrown knots” [Kretschmann, 2010].
Figure 4.1: Intergrown knot [a], encased knot [b]. [Kretschmann, 2010]
“In sections containing knots, most mechanical properties are lower than in clear straight-
grained wood. The reasons for this are:
 The clear wood is displaced by the knot,
 The fibers around the knot are distorted, resulting in cross grain,
 The discontinuity of wood fiber leads to stress concentrations,
 Checking usually occurs around the knots during drying,
 Knots have a greater effect on strength in axial tension than in axial short-column
compression” [Kretschmann, 2010].

“Compared to other building materials, timber demonstrates large variability of the mechanical
properties whereas the variability is recognized between different structural elements and
within single elements. A major reason for the large variability is the presence of knots and
knot clusters in structural timber. Within one knot cluster, knots are growing almost
horizontally in radial direction. Every knot has its origin in the pith. The change of the grain
orientation appears in the area around the knots. In Figure 4.2a the knots [black area] and the
ambient area with deviated grain orientation [grey area] within one cross section of the tree are
illustrated. Since the individual boards are cut out of the natural shape of the timber log during
the sawing process the well-structured natural arrangement of the knots becomes decomposed
due to different sawing patterns. As a result, numerous different knot arrangements appear in
sawn timber [Figure 4.2b, c]” [Kretschmann, 2010].
Figure 4.2: Knot arrangement within the cross section of a tree trunk[a] Influence of
the sawing pattern on the knot distribution within the sawn timber boards [b] and
Resulting knot area within the cross section of one board[c]. [Kretschmann, 2010]
`
Figure 4.3: Notation of the knots. [Kretschmann, 2010]
4.2 Investigation of the Deformation Behavior
4.2.1 Intergrown / Dead Knots
“The growing process of trees and thus, the growing process of branches depend on
environmental conditions. Therefore, the multitude of different branch configurations, affect
the material behavior of solid timber in different ways. From an engineer’s point of view
branches can be subdivided into two groups: Intergrown knots and dead knots. One major

difference between those is the grain orientation around the knots: For dead knots the grains
grow in log direction with a constant distance to the pith; i.e. the grain deviation occurs only in
tangential direction [relative to annual growth ring pattern]. Around intergrown knots grains
are growing in the direction of the log and in direction of the branches; i.e. the grain deviation
occurs in tangential and radial direction. Another difference is that dead knots are surrounded
by bark, contrary to intergrown knots. The transformation from a living branch into a dead
branch occurs within a relatively small time period. Thus, in several cases a knot can be an
intergrown knot on one side and a dead knot on the other side of the timber board. In Figure
4.4 the longitudinal strains [strains in board/load direction], the transversal strains [strains
perpendicular to the board] and the shear strains of the clear wood around an intergrown knot
under a load of 55kN [corresponds to a mean stress within the cross section of 9.92MPa] are
illustrated. The illustration shows significant large longitudinal strains in diagonal direction
[1:30h, 4:30h, 7:30h, and 10:30 h] [1:30h means the direction of the strain, if we assume
the timber board as a clock] and small longitudinal strains in direction 3h and 9h. In load
direction [6h and 12h] the longitudinal strains in the range of zero appear; partly, those are
slightly negative. In all directions the longitudinal strains are decreasing with increasing
distance to the knot. The illustration of the transversal strains presents negative strains in
diagonal direction and positive in direction 3h, 6h, 9h and 12h. The strains in load direction are
clearly greater than those in direction 3h and 9h and have their maximal amount at a distance
of about half knot diameter to the knot. The illustration of the shear strains shows eight
alternating areas with positive and negative strains. The shear strains are decreasing with
increasing distance to the knot” [Gerhard, Jochen &Andrea, 2012].
Figure 4.4: Strain distribution around an intergrown knot under a tensile load of 55kN
[9.92MPa]. [Gerhard, Jochen &Andrea, 2012]

“The estimated strains of the clear wood around dead knots are qualitatively similar to those of
an intergrown knot; i.e. positive longitudinal strains and negative transversal strains in diagonal
direction and alternating positive and negative shear strains. A detailed look on the strains
inside knots and on the adjacent area of knots shows differences between intergrown and dead
knots. Intergrown knots have, in general significant strains inside the knot or rather within the
crack inside the knot [Figure 4.5]; i.e. expansion in longitudinal direction and contraction in
transversal direction. Strain peaks [extension and compression] inside dead knots are usually
in the area of the bark [Figure 4.6]” [Gerhard, Jochen &Andrea, 2012].
Figure 4.5: Strain peaks within an intergrown knot under a tensile load of 55kN
[9.92MPa]. [Gerhard, Jochen &Andrea, 2012]
Figure 4.6: Strain peaks within a dead knot under a tensile load of 66kN [11.9MPa].
[Gerhard, Jochen &Andrea, 2012]
“The strain distribution around and within single centered knots can be explained with a
simplified model [Figure 4.7a]. There, a knot [grey area] and the curved grains around the knot

under tensile load are illustrated. Pulling apart lead to straightening of the grains and thus to a
sidewise pressure on the knot and/or the bark. The bark as well as the crack within the knot
will allow this, whereby negative transversal strain peaks within the knot and positive
transversal strains alongside the knot [direction 3h and 9h] occur. With increasing distance to
the knot [in board direction] the sidewise pressure decreases. With reaching the point of contra
flexure the pressure force turns into tensile force which leads to positive transversal strains. It
is obvious that the magnitude of the transversal strains highly depends on the grain deviation.
In the example described above [Figure 4.4] the maximal amount of the transversal strains is
at a distance of about half knot diameter to the knot. Associated with the positive transversal
strains in the main directions [3h, 6h, 9h and 12h], negative transversal strains in diagonal
direction occur. In longitudinal direction the tensile force leads to significant strains in zones
without tensile resistance, such as cracks perpendicular to the board axis or the bark before and
after the knot. Associated with the local strain peaks the longitudinal strains in the main
directions are relatively small which leads to significantly larger longitudinal strains in
diagonal direction” [Gerhard, Jochen &Andrea, 2012].
Figure 4.7: Simplified model to describe the strain distribution around a single centered
knot [left].and a knot located in the boundary area [right]. [Gerhard, Jochen &Andrea,
2012]
4.2.2 Knots in the Boundary Area
“In this section the deformation behavior of knots, which are located in the boundary area of
the board, is described; e.g. splay knots, narrow side knots and edge knots. One difference

between these knots and knots arranged in the middle of the cross section is that the curved
grains around knots are cut on one side of the board, through to the sawing process.
Figure 4.8: Narrow side knot. [Gerhard, Jochen &Andrea, 2012]
In Figure 4.9 the longitudinal, transversal and shear strains of the narrow side knot illustrated
in Figure 4.8 under a load of 55kN [corresponds to a mean stress within the cross section of
9.92MPa] are illustrated. The dashed line illustrates the knot located on the opposite side of the
board. On the upper side of the board the estimated strains [longitudinal, transversal and shear]
in direction 6h-12h are qualitatively similar to those of a centered single knot [Figure 4.4].
Conspicuous is that the positive transversal strains in direction 3h, 6h and 12h are significantly
larger, compared to those around a single centered knot. On the narrow side of the knot
[direction 3h] positive longitudinal strains, negative transversal strains and negative shear
strains occur. On the bottom side of the board the majority of the extension in longitudinal
direction occurs before and after the knot. Within the area of the knot only marginal strains in
board direction occur. The illustration of the transversal strains shows compression in direction
6h and 12h and extension in direction 3h. The shear strains are positive in direction 3h to 9h
and negative in the opposite direction [9h to 3h]” [Gerhard, Jochen &Andrea, 2012].

Figure 4.9: Strain distribution around a narrow side knot under a load of 55kN
[9.92MPa]. Top: upper side. Bottom: lower side. The dashed line illustrates the knot
located on the opposite side of the board. [Gerhard, Jochen &Andrea, 2012]
“The estimated strains can also be described by a simplified model [Figure 4.7b]. Pulling apart
the timber grains, leads to a sidewise pressure on the knot. On the narrow side the pressure is
significantly smaller then on the opposite side of the knot. That leads to a shift of the knot to
the board edge. Thereby the knot and the grains on the narrow side get pressure and thus
contraction. Furthermore, the shift of the knot leads to an increase of the transversal strains in
direction 6h, 9h and 12h” [Gerhard, Jochen &Andrea, 2012].
4.3 Interaction of Knots
“As described before the natural growing process of trees and the cutting process of timber
boards lead to a countless number of different knot arrangements within one knot cluster. In
the following, the interaction of knots within one knot cluster is analyzed based on their
arrangement. Therefore first, knot clusters containing two knots which are arranged
1) in a row,
2) abreast or
3) diagonal shifted are taken into account.

Analysis of failure in timber boards under tensile loading initiated by knots

Analysis of failure in timber boards under tensile loading initiated by knots

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