project work

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.


Further, the deformation behavior of a knot section containing three knots is described. In a

knot section containing two side knots, which are arranged in a row, both knots show

qualitatively similar deformation behavior then a single knot [Figure 4.4]. Different are only

the shear strains. The illustration shows twelve [instead of eight for a single knot] alternating

areas with positive and negative strains. Between the knots no alternating area occurs. It is

noticed that the positive shear strains of the upper knot and the negative shear strains of the

lower knot are relatively small” [Gerhard, Jochen &Andrea, 2012].

“In both sides of a knot section containing two abreast arranged knots [1 edge knot and 1

narrow side knot] are illustrated. On the upper side the strain fields [longitudinal, transversal

and shear] are similar to those from the upper side of the narrow side knot illustrated in Figure

4.10. Only the shear strains between the knots are different [negative strain]. On the lower side,

the estimated strains are marginal influenced by the narrow side knot” [Gerhard, Jochen

&Andrea, 2012].

“The major part of the strains corresponds to the edge knot. Between the knots significantly

huge transversal strains occur. These results from the both sided pulling of the grain [through

the straightening] in the direction of the knot [Figure 4.7b]. The strains of a knot section

containing two knots which are arranged diagonal shifted are illustrated in Figure 4.12. .In the

illustrated example, there is a relatively huge crack perpendicular to the board axis within the

narrow side knot [on the upper side of the board]. This crack shows a significant opening at a

load of approximately 100kN [18.0MPa]” [Gerhard, Jochen &Andrea, 2012].

“The strains around this knot are highly related to the occurrence of the crack; i.e. no or slightly

negative longitudinal strains and high transversal strains in direction 6h and 12h. The strains

around the side knot, located in the middle of the board, are as usual. On the lower side of the

board only one centered intergrown knot is visible. Around them all strain fields are similar to

the knot described in Section 4.2.1” [Gerhard, Jochen &Andrea, 2012].

“In the last example, the deformation behavior of a knot cluster including three knots [1 side

knot and 2 edge knots] is described Figure 4.13. On the upper side of the board all three knots

show mostly similar deformation behavior like single centered knots. However, between the

knots the strain distributions are partly different or rather ambiguous to describe.

Conspicuously is that for each knot one of the transversal strain peaks in directions 6h or 12h

is always significantly larger than the other. On the lower side of the board similar deformation

behavior is identified as for single side knots and for edge knots, respectively” [Gerhard,

Jochen &Andrea, 2012].


Figure 4.10: Strain distribution around two knots arranged in a row under a tensile

load of 55kN [9.92MPa]. [Gerhard, Jochen &Andrea, 2012]

Figure 4.11: Strain distribution around two knots arranged abreast under a tensile 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]

Two issues are mentionable:

The huge longitudinal strain between the left edge knot and the side knot and

The pronounced shear strains [positive and negative] between the side knot and both

edge knots. It has to be mentioned that on the upper surface of the board no shear strains

between the knots occur” [Gerhard, Jochen &Andrea, 2012].


“The surface deformations of the specimens of the last two examples are evaluated during

nondestructive tension tests [on both sides] and during destructive tension tests [on one side].

Thus Figure 4.12 and Figure 4.13 show the surface strain distribution of the upper and the

lower side of the board under different load. Consequently only a qualitative comparison

between the upper and the lower side can be made. In general the strains identified for single

knots can be applied for knots within a knot cluster. However, in the area between knots the

strain distribution depends clearly on the specific knot cluster; i.e. size and position of each

knot, distance between the knots, arrangement of the knots, and grain deviation around the

knots and so on. It is obvious that the strain field gets more complex with increasing number

of knots” [Gerhard, Jochen &Andrea, 2012].

4.4 Strain History up to Failure “During the destructive tensile test, pictures of the knot cluster are taken until failure of the

timber board. The strains are measured only on one side of the knot section [destructive tensile

test]” [Gerhard, Jochen &Andrea, 2012].

Figure 4.12: Strain distribution around two knots arranged diagonal shifted and the

fracture pattern. Top: upper side, tensile load: 141kN [25.4MPa]. Bottom: lower side,

tensile load: 55kN [9.92MPa]. The dashed line illustrates the knot located on the

opposite side of the board. The dash-dotted line shows the fracture pattern. [Gerhard,

Jochen &Andrea, 2012]


“Therefore the side, with the larger expected strain peaks [in general the side with the greater

amount of knots] is chosen. From each knot cluster the strains are calculated for different loads.

By comparing the strains [at different load levels] the development of the strain distributions

can be detected. In the area of knots or rather within the cracks inside knots, local strain peaks

are detected at relative low levels. With increasing load [tensile force], the strain peaks increase

and strains within the clear wood around the knots occur. Even at a relatively low tensile load

level typical strain distributions arise. In a further increase of load the strains are continuing to

increase, whereas the qualitative pattern of the strains does not change up to the tensile failure.

Knot clusters containing local weak zones outside knots, such as small cracks, show significant

strain peaks [positive or negative] in that zone. Thereby the strain distribution around the weak

zone and thus the strain distribution within the entire knot cluster shows untypical behavior”

[Gerhard, Jochen &Andrea, 2012].

4.5 Correlation between Strains and Failure Behavior “In this section the interaction between the failure behavior and the strains directly before the

occurrence of the failure is analyzed with two examples. The first example [Figure 4.13] shows

a knot cluster containing two knots [1 side knot and 1 narrow side knot]. Herewith the dashed

line illustrates the knot located on the corresponding opposite side of the board. The dash dotted

line shows the fracture pattern. At a tensile load of 141kN the failure occurs. Such a load

corresponds to a mean stress within the cross section of 25.4MPa” [Gerhard, Jochen

&Andrea, 2012].

“For both sides of the boards a good agreement between the fracture pattern and the shear

strains is identified. In this example the fracture is initiated through a crack [perpendicular to

the board axis] within the narrow side knot. On the upper side of the board the fracture pattern

can be separated into four areas:

A. Narrow side knot,

B. Between the knots,

C. Side knot and

D. Between the side knot and the edge of the board.

In area A the fracture is located within the area of the initial crack [significant longitudinal

strains]. In the areas B and D the fracture is located in the boundary area of shear strains. In

area C the fracture occurs around the knot, perpendicular to the grain [in this area no correlation

between the fracture pattern and the strains could be identified]. On the lower side of the board

the fracture pattern can be subdivided into three areas:

A. Left side of the knot,

B. Side knot and


C. Right side of the knot.

In area A and C the fracture is again located in the boundary area of shear strains. In area B the

fracture occurs perpendicular to the grain. In Figure 4.14 the cross section of the failed

specimen is illustrated. Herewith, different types of fracture can be identified. About 10% of

the crack occurs within knots, 30% are shear fracture and 60% are tensile fracture in grain

direction” [Gerhard, Jochen &Andrea, 2012].

Figure 4.13: Strain distribution within a knot cluster containing three knots and the

fracture pattern. Top: upper side, tensile load: 66kN [11.9MPa]. Bottom: lower side,

tensile load: 45kN [8.12MPa]. The dashed line illustrates the knot located on the

opposite side of the board. The dash-dotted line shows the fracture pattern. [Gerhard,


Figure 4.14: Cross section of the failed knot section illustrated in Figure 3.12. [Gerhard,



“The second example shows a knot cluster containing three knots in Figure 4.13. At a tensile

load of 66kN the failure occurs. Such a load corresponds to a mean stress within the cross

section of 11.9 MPa. Between the fracture pattern and the strains only marginal coherence is

identified on the upper side of the board. However, on the lower side the fracture occurs in the

area of significantly huge [positive or negative] shear strains. A detailed look at the fracture

pattern Figure 4.15 shows a shear fracture in the majority of the cross section [~80%] which

explains the relatively low tensile strength. Only in the area between the side knot and one

narrow side knot [on the lower side of the board] a tensile fracture occurs.

In summary, a good agreement between tensile fracture in grain direction and shear strains is

identified. Further, local weak zones, such as small cracks show significant longitudinal and

transversal movements, respectively. In the case of shear failure no permanent local surface

deformation had been detected prior the fracture” [Gerhard, Jochen &Andrea, 2012].

Figure 4.15: Cross section of the failed knot section illustrated in Figure 4.13. [Gerhard,



Chapter 5

Finite Element Modelling of Timber Boards with Knots

The main drawback of timber for structural purposes is its inherent heterogeneity because the

uncertainty of its mechanical behavior and its structural responses requires the use of restrictive

coefficients that drastically reduce the efficacy of this material. These collapse of wooden

structures is mainly due to large displacements, overall instability and brittle failure. The last

cause is, generally, encountered in bolted or nailed joints, where the strength in shear and in

tension perpendicular to the grain represents the weakest point of wood. The analysis of

wooden structures requires an appropriate constitutive law. It is then of importance to deal with

the coupling between the wood behavior in its different orthotropic directions:

1) Ductile compressive behavior with densification perpendicular to the grain.

2) Brittle tensile failure parallel and perpendicular to the grain.

3) Compressive failure with softening parallel to the grain.

From the description provided in chapter 4 it can be seen that knots are the defects that most

reduce the strength of structural size timber; the longitudinal tensile strength is the most

affected property, followed by the flexural strength or modulus of rupture [MOR], the

compressive strength parallel to the grain and the modulus of elasticity [MOE]. [Phillips,

1981]. This reduction in strength is due to a discontinuity in the stress distribution caused by

the knots. In fact, it has been demonstrated for some softwoods, such as Pinus sylvestris [Baño

2009] that under tensile stress conditions, the strength contribution of knots is so low that it is

better to disregard their material contribution by simulating them as holes.

In addition, the fibers located around these defects experience deviations that produce both

shear and perpendicular to the grain stress components which are much more critical than those

stresses parallel to the grain, especially under tensile conditions, where tensile stress

perpendicular to the grain is generated. Therefore, the grain deviation around knots is often the

key aspect for determining the strength of a structural piece.

In the chapter 2 we have discussed about the properties of the timber and also the failure

patterns. It is very important to learn this properties because using timber as structural material,

these properties govern the structural behavior of the timber particles. In the chapter 3 we tried

to make some details about Fracture Mechanics and basic modelling of timber. In the chapter

4 we mainly focused on the drawback of timber structure done by the knot. The knot effects

the timber by reducing its strength both in parallel and perpendicular of the grain. In the above

paragraphs we have discussed about it.


In this chapter, we will try to describe the Finite Element Modelling of the timber on the basis

of cohesive material model. It will focus on brittle failure under tensile loading using cohesive

material model. Cohesive elements have to be generated not only between the knots and

surrounding wood but also along possible crack paths.

5.1 Cohesive zone Model “The idea for the cohesive model is based on the consideration that infinite stresses at the crack

tip are not realistic. Models to overcome this drawback have been introduced by [Dugdale,

1960] and by [Barenblatt, 1962]. Both authors divide d the crack in two parts: One part of the

crack surfaces, region I in Figure 5.1, is stress free, the other part, region II, is loaded by

cohesive stresses. Dugdale introduced the finite stress to be the yield stress, which holds only

for plane stress, but the crack opening stresses can be much higher than the equivalent stress in

a multiaxial stress state. Barenblatt, who investigated the fracture of brittle materials, made

several assumptions about the cohesive stresses: The extension of the cohesive zoned is

constant for a given material [independent from global load] and small compared to other

dimensions. The stresses in the cohesive zone follow a prescribed distribution σ(x), where x is

the ligament coordinate, which is specific for a given material but independent of the global

loading conditions.

Figure 5.1: Dugdale [left] and Barenblatt [right] crack models. [Schwalbe, Scheider,

Cornec, 2012]

Cohesive interface elements are defined between the continuum elements instead, which open

when damage occurs and lose their stiffness at failure so that the continuum elements are

disconnected. For this reason the crack can propagate only along the element boundaries. If the

crack propagation direction is not known in advance, the mesh generation has to make different

crack paths possible.


Figure 5.2: Cohesive model: representation of the physical damage process by

separation function within numerical interfaces of zero height—the cohesive elements.

[Schwalbe, Scheider, Cornec, 2012]

5.1.1 Traction separation law, TSL “The constitutive behavior of the cohesive model is formulated as a traction– separation law

(TSL), which relates the traction, T, to the separation,�, the latter representing the displacement

jump within the cohesive elements. A cohesive element fails when the separation attains a

material specific critical value, ��. The related stress is then zero. The maximum stress reached

in a TSL, the cohesive strength, T0, is a further material parameter” [Schwalbe, Scheider,

Cornec, 2012].

“For a given shape of the TSL, the two parameters, �� and T0, are sufficient for modelling the

complete separation process. In practice, it has been proven useful to use the cohesive

energy,Γ�, instead of the critical separation. The cohesive energy is the work needed to create

a unit area of fracture surface (in fact twice the unit fracture surface because of the two mating

fracture surfaces) and is given by

��= ∫ �(�)��

� [5.1]


Figure 5.3: Form of the TSL: [a] [Needleman, 1987] ,[b] [Needleman, 1990] ,[c]

Hillerborg [1976],[d][ Bazant ,2002], [e] [Scheider, 2001] ,[f] [Tvergaard, 1990]

.[Schwalbe, Scheider, Cornec, 2012]

Beside the form of the curve, which was assumed to be a model quantity, there are two material

parameters, i.e. the maximum separation stress T0, which has to be overcome for final fracture,

and the separation at failure δ0. These quantities define the ’height’ and the ’width’ of the curve,

and give [together with the function of the curve] the dissipated energy per cohesive element

as a result. The damage, in the following denoted as D, is defined as the ratio of the actual to

the maximum separation” [Schwalbe, Scheider, Cornec, 2012].

� = �

�� [5.2]

5.2 Models for brittle failure 5.2.1 Interface elements

“In contrast to the ductile behavior at compression, tensile loading perpendicular to fibers or

shear loading of wood causes cracks and shear planes and results in brittle failure. A continuum

plasticity approach to model the softening behavior of wood at tension failure yields

discretization-dependent result [Zienkiewicz, 2000]. Therefore, regularization is required to

ensure mesh-independent solutions. A possible method to solve this kind of problem is the

use of cohesive elements (interfaces). By using interface-elements, a regularization is not

required because the thickness of the failure zone is predefined by the cohesive elements”

[Schmidt, 2008].


Figure 5.4: Geometry, nodes and local coordinate system of 16-node interface element.

[Schmidt, 2008]

Figure 5.5. Schematic traction-separation behavior of the material model. [Schmidt,

2008]

5.3 Constitutive model for timber under tension and shear “A two-dimensional homogeneous constitutive model for mechanical behavior of timber under

tensile and shear loading is developed in this work. This model covers the following

phenomena:

Elastic and inelastic behavior,

Material orthotropic, both in linear and non-linear range,

Fracture across and along fibers,

Behavior under unloading/reloading is considered to a certain extent,

Only small deformations [lower than 0.05�]. are considered,

The model can be applied both to hardwood and softwood by selecting appropriate

material characteristics.


A constitutive model in general expresses how a material responds to acting stress. In the

present study we idealize timber as a quasi-brittle material. Thus, the material is considered as

a continuum with discontinuities [cracks]. The continuous part is characterized by a stress-

strain law while a traction-separation law describes the behavior of the discontinuities. Fixed

smeared crack model is used to represent the fracture” [Bartůňková, 2013].

5.4 Pure Tension Parallel to Grain From Figure 5.6 to Figure 5.11, several graphs have shown strength properties of timber

Figure 5.6: Course of loading run by strain increments ∆��, ∆��, ∆�� [a]development

of crack opening ��with respect to �� [b]. [Bartůňková, 2013]

Figure 5.7: Stress-strain curve �� − ��[a], �� − �� [b]. [Bartůňková, 2013]


Figure 5.8: Stress-strain curve �� − �� [a], development of non-zero elements of

tangent stiffness matrix �� [b]. [Bartůňková, 2013]

Figure 5.9: Iterated normal traction ��,�� and normal traction function [Bartůňková,

2013]��(�� ) (��,��). [Bartůňková, 2013]

Figure 5.10: Iterated shear traction ��,�� and shear traction function ��(�� , ��

� )

(��,��). [Bartůňková, 2013]


Figure 5.11: Load vs. (clip gauge) displacement for the specimen Ac1. [Bartůňková,

2013]

5.5 Applicability of Model illustrated by Jörg Schmidt The model provided in the Section 5.2 we can see that the cohesive zone model is applicable

for the tension perpendicular to the grain. If we compare to the typical stress strain curve in

Figure 5.12 given in [Henrik, 2013], we can see similarity of the load vs. displacement curve

or the both cases. We can see higher value of strength for perpendicular of grain but lower

value for the parallel to the grain.

Figure 5.12: Approximate range of stress vs. displacement curves for pure mode I (α =

90 ◦) and mode II (α = 0 ◦) loading of Norway spruce. [Henrik, 2013]


We can also compare it to Figure 5.11 which also shows the similar behavior. Now the problem

is that is the model applicable for the knot problem also. Then according to [Bartůňková,

2013], it said

“After full implementation into a finite element code, this model can be used as a basis for the

following advanced approaches to analysis of timber members:

A model that can account for the presence of inhomogeneities such as knots. For this

purpose, the present model would be used to represent clear timber while the

inhomogeneities would be modeled as discrete regions with different properties. This

approach may further require representation of the interface between two materials.”

So, our model and material parameters are applicable for parallel to grain problem. That’s why

we have used this model to simulate behavior of the timber or our project.

5.6 Using ABAQUS for FEM

5.6.1 ABAQUS input for the cohesive element

Material parameters The material parameters used for Norway spruce wood emanate from a literature study

[Schmidt, 2008]. All parameters are mean values.

The elastic moduli’s are

Er = 820 N/mm2,

Et = 430 N/mm2,

El = 13 200 N/mm2,

The shear moduli’s are

Grt = 40 N/mm2,

Gtl = 730 N/mm2,

Grl = 660 N/mm2,

The poison’s ratios are

vrt = 0.24,

vtl = 0.45,

vrl = 0.45.

The strengths are assumed as [Nominal strength]

f t,l = 65.5 N/mm2,

f c, l = −50.3 N/mm2,

f t,r = 3.75 N/mm2,


fc, r = −6.0 N/mm2,

f t,t = 2.79 N/mm2,

fc, t = −6.0 N/mm2,

f v,rt = 1.83 N/mm2,

fv, tl = 5.34 N/mm2,

fv,rl = 6.34 N/mm2

Furthermore, the fracture energies at rt-plane under Mode I and Mode II loading are given by

GI = 280 N/m and GII = 770 N/m.

Mesh in the ABAQUS For the mesh in the ABAQUS these 2d and 3d types of mesh have been generated

Figure 5.13: Mesh of the three-dimensional finite element model. [Moura, 2006]

Figure 5.14: Mesh of the two-dimensional finite element model. [Moura, 2006]

The other procedure done in the ABAQUS are done from several example and exercises found

in the Internet.


5.6.2 ABAQUS Analysis [Timber board without knot] In the first problem we didn’t consider any knot effect in the timber. We consider the board as

pure clear timber board. The force applied here is 50kN. From the Figure 5.15 to Figure 5.18,

the model and the simulated results have shown.

Figure 5.15: Model for simulation in ABAQUS

Figure 5.16: Stress intensity after giving load of 50 kN

Figure 5.17: Stress intensity after giving load of 50 kN [Larger view]


Figure 5.18: Energy vs. time curve after giving load

5.6.2 ABAQUS Analysis [Timber board with knot] In this simulation we have consider a knot in the timber board which can cause reduce of

strength of the timber in the longitudinal direction. We applied a load of 50 kN.

Figure 5.19: Model for simulation in ABAQUS




Figure 5.22: Stress intensity after giving load of 50 kN [Larger view]

Figure 5.23: contact constraint elastic energy [50 kN], energy time curve


5.7 Element library “User defined elements have been developed for two- and three-dimensional models. Since

they behave like contact elements [but contrary to those they may be stressed under tension

loading], they have one dimension less than the surrounding continuum elements. Even though

the cohesive element has no volume in the unloaded and undamaged state, they will be called

3D cohesive element in the following, the elements for two-dimensional problems with be

called 2D cohesive elements, respectively.

The 2D cohesive elements have four nodes with a linear displacement formulation. Elements

are implemented for plane and for axisymmetric problems. There is no difference between

plane stress and plane strain elements, since the stress in the third direction does not influence

the behavior of the cohesive element. The difference between those two element types is the

calculation of the integration point area: While the area of plane elements is calculated by the

half length of the element times its thickness, the integration point area of axisymmetric

elements depends on the distance from the rotation axis. A problem arises when plane stress

elements are used: The thickness of the continuum element cannot be taken into account, since

the actual thickness of the element is not handed to the user element subroutine.

Contrary to the linear formulation of the 2D cohesive elements, the three-dimensional elements

are variable node elements; they can be built using eight nodes and a linear displacement

formulation or with quadratic displacements and 16 or 18 nodes. The stresses are calculated

similar to the 2D cohesive elements at the integration points according to the TSL. Possible

numbers for the integration points per element are 4 or 9. Theoretically any combination of

node and integration point number is possible, but a linear element with nine integration points

is not very useful. A quadratic element with 4 integration points might lead to zero energy

modes but the experience shows that this does not happen in a crack propagation analysis.

Elements are shown in Figure 5.24 In the unreformed state every two nodes share the same

coordinates, but in the drawing they are plotted with a short distance for clarification”

[Scheider, 2001].


Figure 5.24: Element node and integration point positions for the two- and three-

dimensional element. The configuration for the 3D 8 node and 16/18 node elements are

shown separately. [Scheider, 2001]

5.8 Local coordinate system “For the quantitative evaluation of separation of the opposite cohesive surfaces one face is

called upper and other lower surface, but these names are chosen arbitrarily and only used for

the distinction in the text. In the unreformed state both faces are at the same geometrical

position. The separation can be due to sliding or tangential opening” [which is shear fracture],

normal opening [Mode I fracture] or any combination of these two.

There is no difference for the cohesive element, which of either surface is moving absolutely.

From this it follows that neither the upper nor the lower face can be the reference face for the

coordinate system. Instead of these two other possibilities for the definition of the local

coordinate system are implemented:

1. The original position of the cohesive element is taken as the reference for the coordinate

system. This is analogous to the theory of small deformations.

2. Analogous to the theory of large deformation the coordinate system is moving with the

element using a mid-section face, that is as the bisector between upper and lower surface”

[Scheider, 2001].

Figure 5.25: Definition of the local coordinate system within the cohesive element.

[Scheider, 2001]


5.9 Result In our project we didn’t perform any test. And also due to technical problem we didn’t manage

to get suitable data or analysis. But similar type of test was done by [Jan, Anders, Bertil,

2010]. For analysis of data and finding the result we have demonstrated their test result and

simulation work. In the chapter 4 we also have provided test results on timber boards with

knots. These results also show the strength properties of timber differ depending on types and

position of the knots.

“The results of the experiments have been compared with those obtained from FE simulations

using the ABAQUS, FE software. In the FE calculations, a 3D linear elastic model of the test

specimen’s behavior under loading was used. The model comprised about 113000

elements and 483000 nodes resulting in 1449000 degrees of freedom. In the simulations, a

tension load of 30 kN was applied as a distributed load over the end surfaces of

the specimen” [Jan, Anders, Bertil, 2010].

“The material data needed for the numerical calculations, expressed in terms of elastic

constants for spruce with moisture ratio of 12 %.The constants used for both clear wood and

knot were EL = 13500, ER = 893, ET = 481, GLR= 716, GLT=500 and GRT =29 MPa for the

moduli of elasticity and the shear moduli, and υLR = 0.43, υLT =0.53 and υRT = 0.42 for Poisson’s

ratios. The indices L, R and T refer tithe longitudinal, radial and tangential directions,

respectively, of the modelled orthotropic wood material. The longitudinal direction of the clear

wood was oriented parallel with the longitudinal direction of the specimen, whereas the

longitudinal direction of the knot was oriented perpendicular to the mentioned clear wood

direction” [Jan, Anders, Bertil, 2010].


Figure 5.26: Longitudinal surface strains (εy) for maximum load 30 kN, recorded for

Load tests no. A1-A4. Top row: Surface photos. Middle row: εy contour plots. Bottom

row: εy section diagrams for the sections (dashed lines) shown inthe middle row. [Jan,

Anders, Bertil, 2010]


Figure 5.27: Lateral surface strains (εx) for maximum load 30 kN, recorded for Load

tests no A1-A4. Top row: Surface photos. Middle row: εx contour plots. Bottom row: εx

section diagrams for the sections (dashed lines) shown in the middle row. [Jan, Anders,

Bertil, 2010]


Figure 5.28: Results from FE simulation: Top row: Longitudinal strains and FE mesh.

Bottom row: Lateral strains. [Jan, Anders, Bertil, 2010]


Chapter 6

Conclusion

6.1 Limitation The work done in this project are mostly from literature work. That’s why all information

needed for description of every aspect cannot be provided. So, the data needed for simulation

and their details is not included in this project because it will make this paper bulkier.

The most of the test and simulation done for tension properties are for the perpendicular to the

grain. So, data available for parallel to grain are so little that with that information it is too hard

to reach a decision. A detailed test and a detailed simulation is necessary for the understating

of this property. In the timber, for tension most failures occur at the perpendicular to the grain.

For that, most researches have done on that. But for increasing the strength of timber it’s really

needed to do research on parallel to grain properties. Because, recently the applicability of

timber boards have increased a lot and it’s always creating new dimension of structural timber.

6.2 Proposals for future work The result found in the simulation was not optimum level, because of the limitation of research

paper on parallel to grain properties. And also the meshing done in our simulation cannot give

a real picture how this strain is progressing and how the strain flow is transferring through the

grain .That’s why we need to use Streamline meshing,

“For modelling failure initiated by knots, plasticity formulations and cohesive elements are

applied. It is shown, that the regular mesh works well in combination with plasticity.

However, to simulate brittle failure at tensile loading cohesive elements should be applied. To

distinguish knots and surrounding wood, an automatic meshing procedure for timber boards is

introduced. This Stream Line meshing is restricted to boards without splay knots located far

from pith until know. The development of the SL-meshing is work in progress. By using

equipotential lines similar to the stream lines, meshing in transverse direction can be improved.

By using quadratic shape functions with 20-node-elements, streamline curves and knot shapes

could be modelled more accurately. In addition, a suitable adaptive meshing procedure will be

developed to simulate brittle failure” [Jenkel, 2014].


Figure 6.1: Stream line approach: (a) stream lines around knots, velocity vectors in (b)

regular mesh and (c) direct meshing. [Jenkel, 2014]

Figure 6.2: SL-mesh: (a) determination of crack path, (b) simulation results for timber

board. [Jenkel, 2014]

“Based on this SL-mesh, cohesive elements can be generated in between knots and surrounding

wood and at the positions where cracking occurs. The method is illustrated for a single knot in

a board under tensile loading in Fig. 12(a). In an iterative simulation, the TSAI-WU criterion

is utilized again to generate cohesive elements in-between continuum elements. In Figure

6.2(a), the elements, where the criterion is fulfilled, are highlighted in black and gray whereas

the darker shade represents more failed integration points inside an element. If the criterion is

fulfilled in an iteration step, the corresponding stresses are checked. Only if failure is caused

by tensile and shear stress, a cohesive element is generated in-between the failed elements. The

direction is determined by the corresponding stresses as well. When a complete crack path is

derived, a final simulation with the new models carried out, whereas brittle failure can be

described solely by the cohesive elements” [Jenkel, 2014].


REFERENCES

[Anderson, 2005]. Anderson, T. L. [2005], Fracture Mechanics - Fundamentals and

Applications, ISBN 978-1-4200- 5821 -5 [eBook – PDF].

[Baño 2009]. Baño V [2009]. Numerical analysis of wood strength loss due to the presence of

knots. Dissertation [in Spanish], University of Santiago de Compostela.

[Barenblatt, 1962]. Barnblatt, G. [1962]. The mathematical theory of equilibrium c racks in

brittle fracture. Advances in Applied Mechanics, 7:55–129.

[Bartůňková, 2013]. Eliška Bartůňková, [2013], Method for determination of the softening

behavior of wood and the applicability of a nonlinear fracture mechanics model, CTU in

Prague.

[Boone, Wawrzynek, 1987]. Boone, Wawrzynek TJ, Wawrzynek PA, Ingraffea AR [1987].

Finite element modelling of fracture propagation in orthotropic materials. Eng Fract Mech

26[2]:185–201.

[Bostrom, 1992]. Bostrom, L, [1992], Method for determination of the softening behavior of

wood and the applicability of a nonlinear fracture mechanics model, Doctoral Thesis, Report

TVBM-1012, Lund, Sweden.

[Bowie and Freese 1972]. Bowie OL, Freese CE [1972]. Central crack in plane orthotropic

rectangular sheet. Int J Fract Mech 8[1]:49–58.

[Cook and Gordon 1964]. Cook J and Gordon JE [1964]. A mechanism for the control of

crack propagation in all brittle systems. Proceedings of the Royal Society of London, A, 282,

508.

[Curtin and Scher, 1990]. Curtin W, Scher H [1990]. Brittle fracture of disordered materials:

a spring network model.J Mater Res 5[3].535–553.

[Davids et al. 2003]. Davids WG, Landis EN, Vasic S [2003]. Lattice models for the prediction

of load-induced failure and damage in wood. Wood Fiber Sci 35[1]:120–135.

[Dinwoodie, 1981]. Dinwoodie, J. M. [1981], Timber, Its Nature and Behavior, New York,

ISBN 0-419-25550-8.

[Dugdale, 1960]. Dugdale, D. S. [1960]. Yielding of steel sheets containing slits. J. Mech.

Phys.Solids, 8:100–104.

[Feenstra, 1993]. Feenstra, P. H. [1993], Computational aspects of biaxial stress in plain and

reinforced concrete,Dissertation, Delft University of Technology, Delft, The Netherlands.


[Gerhard, Jochen &Andrea, 2012]. Gerhard Fink, Jochen Kohler, Andrea Frangi, [2012],

Experimental analysis of the deformation and failure behavior of significant knot clusters.

[Gibson and Ashby, 1988]. Gibson, L. J. Ashby, M, F, [1988]. Cellular solids, Structure and

Properties, Oxford: Pergamoon.

[Henrik, 2013]. Henrik Danielsson. [2013]. Perpendicular to grain fracture analysis of wooden

structural elements models and applications, doctoral Thesis. Division of Structural Mechanics,

LTH, Lund University, Box 118, SE-221 00 Lund, Sweden.

[Herrmann and Roux, 1990]. Herrmann HJ, Roux S [1990]. Statistical models for the fracture

of disordered media. North Holland, Amsterdam.

Hillerborg [1976]. Hillerborg, A, Modeér, M, Petersson, P.E. [1976]. Analysis of crack

formation and crack growth inconcrete by means of fracture mechanics and finite elements.

Cem. Concr. Res. 6, 773–782.

[Holmberg, 1998]. Holmberg, S. Persson, K. Petersson, H. [1998], Nonlinear mechanical

behavior and analysis of wood and fiber materials, Division of Structural Mechanics, Lund

University, Lund, Sweden.

[Christian, 2014]. Christian Jenkel, Michael Kaliske. [2014].Finite element analysis of timber

containing branches – An approach to model the grain course and the influence on the structural

behavior.

[Jan, Anders, Bertil, 2010] Jan Oscarsson, Anders Olsson, Bertil Inquest. [2010]. Strain fields

around a traversing edge knot in a spruce specimen exposed to tensile forces.

[Jirásek, 2012]. Jirásek, M. [2012], Modelling of localized inelastic deformation, lecture notes,

CTU in Prague.

[Jirasek and Bazant, 1995]. Jirasek M, Bazant Z [1995]. Macroscopic fracture characteristics

of random particle systems.Int J Fract 69:201–228.

[Kanninen et al. 1977]. Kanninen MF, Rybicki EF, Brinson HF [1977]. A critical look at

current applications of fracture mechanics to the failure of fiber-reinforced composites.

Composites 8:17–22.

[Kretschmann, 2010]. Kretschmann, D. E. [2010], Mechanical Properties of Wood, Chapter

5, Wood handbook - Wood as an engineering material, General Technical Report FPL-GTR-

190, Madison, WI: U.S. Department of Agriculture, Forest Service, Forest Products

Laboratory, 508 p.


[Kuklík, 2008]. Kuklík, P. [2008], Fire Resistance of Timber Structures, Handbook 1 – Timber

Structures, Chapter 17, Leonardo da Vinci Pilot Project, CZ/06/B/F/PP/168007, Educational

Materials for Designing and Testing of Timber Structures – TEMTIS.

[Mallory, 1987]. Marcia Patton-Mallory Steven, M. Cramer. [1987]. Fracture mechanics: a

tool for predicting wood component strength, FOREST PRODUCTS JOURNAL Vol. 37. No.

7/8.

[Mark, 1967]. Mark RE [1967]. Cell Wall Mechanics of Tracheids, Yale University Press,

New Haven.

[Moura, 2006]. M.F.S.F. de Moura, M.A.L. Silva, A.B. de Morais, J.J.L. Morais. [2006].

Equivalent crack based mode II fracture characterization of wood.

[Needleman, 1990]. Needleman, A. [1990]. An analysis of decohesion along an imperfect

interface. International Journal of Fracture, 42:21–40.

[Needleman, 1987]. Needleman, A. [1987]: A continuum model for void nucleation by

inclusion debonding. J. Appl.Mech. 54, 525–531.

[Oudjene and Khelifa 2009]. Oudjene, M. Khelifa, M. [2009], Finite element modelling of

wooden structures at large deformations and brittle failure prediction, Materials and Design,

Nancy-Universite, Laboratoired’Etudes et de Recherche sur le Matériau Bois.

[Peter, 2010]. Peter Domone, John Illston, [2010]. Construction Materials: Their Nature and

Behavior, Fourth Edition, Chapter 54: Strength and failure in timber.

[Phillips, 1981]. Phillips GE, Bodig J, Goodman JR [1981]. Flow grain analogy. Wood Sci

14:55–64.

[Reinhardt, 1986]. Reinhardt H. W. Cornelissen, H. A. W. Hordijk, D. A. [1986], Tensile tests

and failure analysis of concrete. Journal of Structural Engineering, ASCE, 112:2462–2477.

[Reiterer, 2002]. Reiterer, A. Sinn, G. Stanzl-Tschegg, S. E. [2002], Fracture characteristics

of different wood species under mode I loading perpendicular to the grain. Mater Sci Eng

A332:29–36.

[Ritter, 1990]. Michael A. Ritter, [1990]. Timber Bridges - Design, Construction, Inspection,

and Maintenanceloose4 leaf]. Loose Leaf – and from “Chapter 3: Properties of Wood and

Structural Wood Products”.

[Scheider, 2001]. Scheider, I. [2001]. Cohesive model for crack propagation analyses of

structures with elastic–plastic material behavior, GKSS research center Geesthacht, Dept.

WMS.


[Schmidt, 2008]. Jörg Schmidt, Michael Kaliske. [2008]. Models for numerical failure

analysis of wooden structures, Technische Universität Dresden, Institute for Structural

Analysis, D-01062 Dresden, Germany.

[Schlangen and Garboczi, 1996]. Schlangen E, Garboczi E [1996]. New method for

simulating fracture using an elastically uniform random geometry lattice. Int J Eng Sci

34[10]:1131–1144.

[Schlangen, 1995]. Schlangen E [1995]. Experimental and numerical analysis of fracture

processes in concrete. Research report, Technical University of Delft, Delft, The Netherlands.

[Schlangen and Garboczi, 1997]. Schlangen E, Garboczi E [1997]. Fracture simulations of

concrete using lattice models: computational aspects. Eng Fract Mech 57[2/3]:319–332.

[Schwalbe, Scheider, Cornec, 2012]. Karl-Heinz Schwalbe, Ingo Scheider, Alfred

Cornec.[2012] Guidelines for Applying Cohesive Models to the Damage Behaviour of

Engineering Materials and Structures, ISBN 978 3 642 29493 8.

[Schwalbe and Cornec, 1994]. Schwalbe, K.-H. and Cornec, A. [1994]. Modelling crack

growth using local process zones.Technical report, GKSS research centre, Geesthacht.

[Sihet, 1965]. Sihet GC, Paris PC, Irwin GR [1965]. On cracks in rectilinear anisotropic bodies.

Int J Fract1:189–203.

[Smith et al. 2003]. Smith I, Vasic S [2003]. Fracture behaviour of softwood. Mech Mater

35:803–815.

[Tvergaard, 1990]. Tvergaard, V. [1990]. Material failure by void growth to coalescence,

pages 83–151. Academic Press.7.

[Vasic, Smith and Landis, 2004]. Svetlan Vasic, Ian Smith, Eric Landis, Finite element

techniques and models for wood fracture mechanics Wood Sci Technol [2005]. 39: 3–17 DOI

10.1007/s00226-004-0255-3.

[Vasic 2000]. Vasic S [2000]. Applications of fracture mechanics to wood. PhD Thesis,

University of New Brunswick, Fredericton, NB, Canada.

[Vasic and Smith 2002]. Vasic S, Smith I [2002]. Bridging crack model for fracture of spruce.

Eng Fract Mech 69:745– 760.

[Wernersson 1990]. Wernersson H [1990]. Fracture characterization of wood adhesive joints.

Report TVSM-1006, Lund University, Division of Structural Mechanics, Lund, Sweden.

[Wiedenhoeft, 2010]. Wiedenhoeft, A, [2010], Structure and Function of Wood, Chapter 3,

Wood handbook - Wood as an engineering material, General Technical Report FPL-GTR-190,


Madison, WI: U.S. Department of Agriculture, Forest Service, Forest Products Laboratory, 508

p.

[Wu, 1967]. Wu, E.M. [1967]. Application of fracture mechanics to anisotropic plates. Am.

Soc. Mech. Eng. J. Appl. Mech. 34[4]:967-974.

[Zienkiewicz, 2000]. Zienkiewicz O, Taylor R. [2000].The finite element method — Volume

2: Solid mechanics. Oxford: Butterworth-Heinemann.

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