Evaluation of Splitting Capacity of Bottom Rails in Partially Anchored Timber Frame Shear Walls

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DOCTORA L T H E S I S

Department of Civil, Environmental and Natural Resources Engineering Division of Structural and Construction Engineering

Evaluation of Splitting Capacity of Bottom Rails in Partially Anchored Timber Frame

Shear Walls

Giuseppe Caprolu

ISSN 1402-1544 ISBN 978-91-7583-149-7 (print)

ISBN 978-91-7583-150-3 (pdf) Luleå University of Technology 2014

Giuseppe Capr olu Ev aluation of Splitting Capacity of Bottom Rails in P ar tially Anchor ed Timber Frame Shear W alls

ISSN: 1402-1544 ISBN 978-91-7583-XXX-X Se i listan och fyll i siffror där kryssen är

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Evaluation of Splitting Capacity of Bottom Rails in Partially Anchored Timber Frame

Shear Walls

Giuseppe Caprolu

Luleå University of Technology

Department of Civil, Environmental and Natural Resources Engineering

Division of Structural and Construction Engineering –

Timber Structures

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Printed by Luleå University of Technology, Graphic Production 2014 ISSN 1402-1544

ISBN 978-91-7583-149-7 (print) ISBN 978-91-7583-150-3 (pdf) Luleå 2014

www.ltu.se

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Abstract I

ABSTRACT

The horizontal stabilization of timber frame buildings is often provided by shear walls. Plastic design methods can be used to determine the load-carrying capacity of fully and partially anchored shear walls. In order to use these methods, a ductile behaviour of the sheathing-to-framing joint must be ensured. If hold-downs are not provided, the vertical uplifting forces are transferred to the substrate by the fasteners of the sheathing-to-framing joints. Since the forces in the anchor bolts and the sheathing-to-framing joints do not act in the same vertical plane, the bottom rail will be subjected to bending in the crosswise direction, and splitting of the bottom rail may occur. If the bottom rail splits the applicability of the plastic design method for partially anchored shear walls is questionable. This doctoral thesis addresses the problem of brittle failure of the bottom rail in partially anchored timber frame shear walls.

The first part of the study comprised of two basic experimental programs, for single-sided and double-sided sheathed shear walls. The aim was to evaluate the different failure modes and the corresponding splitting capacity of the bottom rail. Two brittle failure modes were observed: (1) a crack opening from the bottom surface of the bottom rail; and (2) a crack opening from the side surface of the bottom rail along the line of the fasteners of the sheathing-to-framing joints. It was found that the distance between the washer edge and the loaded edge of the bottom rail has a decisive influence on the type of failure mode and the maximum failure load of the bottom rail.

Two theoretical models for the load-carrying capacity for each type of failure mode based on a fracture mechanics approach are studied and validated. The two analytical closed-form solutions are in good agreement with the test results. The fracture mechanics models seem to capture the essential behaviour and to include the decisive parameters of the bottom rail. These parameters can easily be determined and the fracture mechanics models can be used in design equations for bottom rails in partially anchored shear walls. Also, an extended fracture mechanics model for the load-carrying capacity for each type of failure mode is presented and evaluated.

The present study discusses the splitting behaviour of the bottom rail

and provides methods to determine the splitting capacity for two brittle

failure modes, splitting of the bottom surface (mode 1) and of the side

surface of the rail (mode 2). By these means brittle failure of the

bottom rail can be avoided and the full plastic load-carrying capacity of

the sheathing-to-framing joints can be utilized.

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Sammanfattning III SAMMANFATTNING

Horisontalstabiliseringen av byggnader med trästomme sker ofta via skivverkan. Plastiska dimensioneringsmetoder kan användas för att bestämma bärförmågan för fullt och partiellt förankrade skjuvväggar.

För att kunna använda dessa metoder, måste ett duktilt beteende hos förbandet mellan skiva och stomme säkerställas. Om förankringsjärn inte används, kommer de vertikala lyftkrafterna att överföras till underlaget via förbindare mellan skiva och stomme. Eftersom krafterna i förankringsbultarna och förbindarna mellan skiva och stomme inte verkar i samma vertikala plan kommer syllen att utsättas för böjning vinkelrätt fibrerna och uppsprickning av syllen kan resultera. Om syllen spricker är det tveksamt om en plastisk dimensioneringsmetod kan användas för partiellt förankrade skjuvväggar.

Den första delen i studien innehöll två experimentella delstudier, en för enkelsidig och en för dubbelsidiga skivor. Syftet var att utvärdera olika brottmoder och tillhörande kapacitet för syllen. Två spröda brottmoder observerades: (1) en spricka längs syllen öppnas från botten på syllen och uppåt och (2) en spricka längs syllen öppnas från sidan på syllen och propagerar i huvudsak horisontellt längs förbindarna mellan skiva och stomme. Avståndet mellan brickans kant och den belastade änden av syllen har en avgörande påverkan på brottmod och maximal last för syllen.

Två teoretiska modeller för bärförmågan för varje brottmod har härletts, båda baserade på brottmekanik. De två analytiska lösningarna överensstämmer väl med testresultaten. De brottmekaniska modellerna fångar det grundläggande beteendet hos syllen och innehåller de avgörande parametrarna. Dessa parametrar kan enkelt bestämmas och brottmekaniska modeller kan användas i dimensioneringssituationen av syllen i partiellt förankrade skjuvväggar. En vidareutveckling av de brottmekaniska modellerna med förfinad modellering presenteras och utvärderas också.

Studien diskuterar uppsprickning av syllen och visar på metoder för

att bestämma bärförmågan för två spröda brott: uppsprickning av

undersidan på syllen (mod 1) och av sidan på syllen (mod 2). Genom

att använda metoderna kan spröda brott i syllen undvikas och full

plastisk bärförmåga hos förbanden mellan skiva och stomme utnyttjas.

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Acknowledgements V

ACKNOWLEDGEMENTS

First of all I would like to express my sincere gratitude to my supervisors, Professor Ulf Arne Girhammar and Associate Professor Helena Lidelöw for their support during these five years. I would also like to thanks Professor Bo Källsner for sharing his broad knowledge in timber structures and to Professor Barbara De Nicolo and Professor Massimo Fragiacomo for helping me to start this journey.

Many thanks to all my colleagues at the University, for all I learned from them and for their help. I would like also to thank the staff working at the laboratory at Umeå University and SP laboratory in Stockholm, where I performed all my experimental studies.

I take this chance to thank the Sardinian Region for its financial support with the PhD scholarship program “Master and Back” that gave me the idea and possibility to do this experience.

Finally I would like to thank my family for their mental support and all friends and people I have met during these five years, you are too many to be mentioned one by one, but I have to mention my best friends Nicola and Damiano, you made my stay in cold Luleå warmer.

Giuseppe Caprolu

Luleå, November 2014

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List of publications VII LIST OF PUBLICATIONS

The thesis is based on studies presented in the following publications:

I. Caprolu G., Girhammar U. A., Källsner B. and Lidelöw H.

(2014) Splitting capacity of bottom rail in partially anchored timber frame shear walls with single-sided sheathing. Published in The IES Journal Part A: Civil & Structural Engineering, 7:83 – 105.

II. Caprolu G., Girhammar U. A. and Källsner B. (2014) Splitting capacity of bottom rail in partially anchored timber frame shear walls with double-sided sheathing. Published online in The IES Journal Part A: Civil & Structural Engineering, November 2014.

III. Caprolu G., Girhammar U. A. and Källsner B. (2014) Analytical models for splitting capacity of bottom rails in partially anchored timber frame shear walls based on fracture mechanics. Submitted to Engineering Structures in November 2014.

IV. Jensen J. L., Caprolu G. and Girhammar U.A. (2014) Fracture mechanics models for brittle failure of bottom rails due to uplift in timber frame shear walls. Submitted to Structural Engineering and Mechanics in November 2014.

V. Caprolu G., Girhammar U. A. and Källsner B. (2014) Comparison of models and tests on bottom rails in timber frame shear walls experiencing uplift. Submitted to Material and Structures in November 2014.

In addition to the publications listed above, conference contributions have been written during the project:

x Caprolu G., Girhammar U. A., Källsner B. and Johnsson H.

(2012) Tests on splitting failure capacity of the bottom rail due to uplift in partially anchored shear walls. In Proceedings of the 12

^th

World Conference on Timber Engineering, Auckland, New Zealand.

x Caprolu G., Girhammar U. A., Källsner B. and Vessby J. (2012)

Analytical and experimental evaluation of the capacity of the

bottom rail in partially anchored timber frame shear walls. In

Proceedings of the 12

^th

World Conference on Timber

Engineering, Auckland, New Zealand.

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Table of contents IX TABLE OF CONTENTS

ABSTRACT ... I SAMMANFATTNING ... III ACKNOWLEDGEMENTS ... V LIST OF PUBLICATIONS ... VII PART I ... XI

NOTATIONS AND SYMBOLS ... 1

1 INTRODUCTION ... 3

1.1 B

ACKGROUND

... 3

1.2 A

IMS AND SCOPE

... 7

1.3 L

IMITATIONS

... 8

1.4 O

UTLINE OF THE THESIS

... 8

2 THEORETICAL CHAPTER ... 11

2.1 M

ODELLING OF SHEAR WALLS

... 11

2.1.1 Elastic models ... 13

2.1.2 Finite element models ... 17

2.1.3 Plastic models ... 18

2.1.4 Design method according to Eurocode 5... 20

2.2 F

RACTURE MECHANICS

... 21

2.2.1 Strain energy release rate ... 23

3 EXPERIMENTAL STUDIES... 27

3.1 S

PLITTING CAPACITY OF BOTTOM RAIL

... 27

3.1.1 Material properties ... 27

3.1.2 Test programmes ... 27

3.1.3 Test set-up ... 28

3.2 M

ATCHING TESTS OF BRITTLE FAILURE OF BOTTOM RAIL

,

FRACTURE ENERGY AND TENSILE STRENGTH PERPENDICULAR TO THE GRAIN

... 30

3.2.1 Bottom rail experimental program ... 32

3.2.2 Fracture energy ... 33

3.2.2.1 Material properties ... 33

3.2.2.2 Test program ... 33

3.2.2.3 Test set-up ... 34

3.2.3 Tensile strength perpendicular to the grain ... 35

3.2.3.1 Material properties ... 35

3.2.3.2 Test program ... 35

3.2.3.3 Test set-up ... 35

4 ANALYTICAL MODELS ... 39

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X Table of contents

4.1 F

AILURE MODE

1 ... 39

4.2 F

AILURE MODE

2 ... 41

5 RESULTS ... 43

5.1 B

OTTOM RAIL TEST RESULTS

... 43

5.1.1 Failure modes ... 43

5.1.2 Load-time curves and crack development ... 46

5.1.3 Failure loads ... 48

5.2 M

ATCHING TESTS OF BRITTLE FAILURE OF BOTTOM RAIL

,

FRACTURE ENERGY AND TENSILE STRENGTH PERPENDICULAR TO THE GRAIN

... 57

5.2.1 Bottom rail ... 57

5.2.2 Fracture energy ... 58

5.2.3 Tensile strength perpendicular to the grain ... 60

6 ANALYSIS AND DISCUSSION ... 61

6.1 B

OTTOM RAIL EXPERIMENTAL PROGRAMMES

... 61

6.1.1 Distance s ... 61

6.1.2 Pith orientation ... 62

6.2 B

OTTOM RAIL ANALYTICAL MODELS

... 63

7 CONCLUSIONS ... 69

8 FUTURE WORK ... 73

REFERENCES ... 75 PART II Appended papers

PAPER I

PAPER II

PAPER III

PAPER IV

PAPER V

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PART I

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Notation and symbols 1 NOTATIONS AND SYMBOLS

A area of the crack [mm

²

] C compliance [mm/N]

DOF degree of freedom DS double-sided

E modulus of elasticity [MPa]

FEM finite element method G shear modulus [MPa]

G

c

critical fracture energy [N/m]

G

f

fracture energy [N/m]

LEFM linear elastic fracture mechanics

LR longitudinal-radial crack orientation LT longitudinal-tangential crack orientation NLFM nonlinear fracture mechanics

Ø diameter [mm]

P bottom rail failure load [kN]

P

_u

failure load of a loaded elastic body [kN]

PD pith downwards PU pith upwards R radial direction

R

²

coefficient of determination RL radial-longitudinal crack orientation RMSE root mean square error

RT radial-tangential crack orientation SS single-sided

T tangential direction

TL tangential-longitudinal crack orientation TR tangential-radial crack orientation XFEM extended finite element method a crack length [mm]

b width of the bottom rail [mm]

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2 Notations and symbols

b

_crack1

distance between a vertical crack and the loaded edge of the bottom rail [mm]

b

_crack2

length of a horizontal crack before change to the vertical direction [mm]

b

_e

“cantilever span” for the geometry used to derive formulas for failure mode 1 [mm]

c additional cantilever length [mm]

d thickness of the fracture energy specimen [mm]

e depth of the tensile strength perpendicular to the grain specimen [mm]

f

_t,90

tensile strength perpendicular to the grain [MPa]

h depth of the bottom rail [mm]

h

_c

distance between the notch and the upper edge of the fracture energy specimen [mm]

h

_e

depth of the “cantilever beam” used to derive formulas for failure mode 1 [mm]

l length of the bottom rail [mm]

s distance between the edge of the washer and the loaded edge of the bottom rail [mm]

t depth of the fracture energy specimen [mm]

u width of the tensile strength perpendicular to the grain specimen [mm]

v thickness of the tensile strength perpendicular to the grain specimen [mm]

į deflection of the loading point [mm]

į

_b

contribution from bending to the deflection of the loading point [mm]

į

_r

contribution from shear to the deflection of the loading point [mm]

į

_v

contribution from rotation to the deflection of the loading

point [mm]

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Introduction 3 1 INTRODUCTION

This chapter outlines the motivation for this thesis followed by the aim of the research, its limitations and outline of the thesis structure.

1.1 Background

Timber frame building systems are a commonly used solution in timber housing construction. Timber frame buildings are made up by a frame of timber joists and studs, sheathed with panels joined to the wood elements. Wood-based panels, such as plywood, OSB, fibre- board or chipboard, are commonly used in timber frame buildings.

Gypsum panels or similar products are also widely used in combination with timber, mainly to provide fire resistance. The timber frame concept is also competitive for multi-storey and multi-residential buildings (Thelandersson and Larsen 2003). In Figure 1.1 examples of multi-storey timber frame house are shown.

a) b)

Figure 1.1 Examples of multi-storey timber frame buildings built in Stockholm

(Sweden): (a) 2011; and (b) 2009. (Lindbäcks Bygg).

One of the main issues to ensure when designing timber frame

buildings is the horizontal stability. Since timber structures are light-

weight, due to the high strength to weight ratio of wood, actions of

horizontal force as wind and earthquake can cause high load

concentrations and large deformations in timber structures. With

increasing number of storeys the issue becomes more severe, as the

self-weight of the structure is not sufficient to provide the necessary

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4 Introduction

stabilising force to counteract overturning, Thelandersson and Larsen (2003).

The stabilisation of timber frame building is often provided by shear walls. Shear walls are structural elements designed to transmit forces in its own plane. They carry wind or other horizontal forces, called racking loads, in the plane of the wall (shear loads) in addition to the vertical loads and lateral pressure on their surface. They are composed of a frame made of vertical elements, studs, connected to two horizontal elements, top and bottom rail, and sheathed with panels. In Figure 1.2 the behaviour of a shear wall subjected to wind load and its typical construction details are shown.

Figure 1.2 Typical shear wall behaviour: (a) the building is loaded by wind load

and one half of the total wind load is transferred to the roof level; (b) the roof diaphragm, acting as a deep horizontal beam, transmits the load to the shear wall;

(c) the shear wall transfers the load to the foundation; and (d) construction details

of the shear wall structure.

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Introduction 5 The lateral wall, Figure 1.2a, is considered to be simply supported at roof and foundation, transferring one half of the total wind load to the roof level. Then the roof diaphragm, acting as a deep horizontal beam, transmits the load to the shear wall, Figure 1.2b. In turn, the shear wall transfers the load to the foundation, Figure 1.2c. The structural behaviour of shear walls is to a large extent determined by the sheathing-to-framing joints and by the connection between walls and the surrounding structure. Of particular importance is the anchoring of the shear wall to the floor/foundation. Sometimes tie-down devices are used for anchorage of the end studs of the shear wall. On other occasions only the bottom rail is anchored to the floor foundation, Källsner and Girhammar (2009).

As pointed out by Prion and Lam (2003) it is important to understand the difference in the anchorage systems: anchor bolts and hold-downs, Figure 1.3.

a) b)

Figure 1.3 Different ways to anchor a shear wall: (a) the anchor bolt provides

horizontal shear continuously between the bottom rail and the foundation; and (b) the hold-down serves as a vertical anchorage device between the leading stud and the foundation.

Anchor bolts provide horizontal shear continuity between the bottom rail and the foundation. Hold-downs serve as vertical anchorage devices between the vertical end studs and the foundation.

In fully anchored shear walls, where both of them are provided, the

vertical loads are directly transferred to the substrate, resulting in a

concentrated force at the end of the wall, as shown in Figure 1.4a. The

notation fully anchored means that the bottom rail fully interacts with

the substrate and that there is no uplift of the studs of the walls,

especially of the leading stud. When hold-downs are not provided, in

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6 Introduction

partially anchored shear walls, the corresponding tying-down forces may be replaced by vertical loads from dead-weight or anchorage forces transferred from transverse walls. The bottom row of nails transmits the vertical forces in the sheathing to the bottom rail (instead of the vertical stud) where the anchor bolts will further transmit the forces to the foundation. This results in a distributed force, as shown in Figure 1.4b.

Figure 1.4 Two principal ways to anchor timber frame shear walls subjected to

horizontal loading: (a) fully anchored shear wall – concentrated anchorage of the leading stud, i.e. using a hold-down; and (b) partially anchored shear walls – distributed anchorage of the bottom rail through the sheathing-to-framing joints.

Since the forces in the anchor bolts and the sheathing-to-framing joints do not act in the same vertical plane, the bottom rail will be subjected to crosswise bending and shear, and splitting of the bottom rail may occur, as shown in Figure 1.5.

a) b)

Figure 1.5 Examples of splitting failure of the bottom rail in partially anchored

timber frame shear walls: (a) splitting failure along the bottom side of the rail; and (b) splitting failure along the edge side of the rail. For both cases the left pictures refers to a bottom rail with single-sided sheathing and the right pictures to a bottom rail with double-sided sheathing.

Nowadays, in Europe, two design methods of shear walls exist.

They are given in Eurocode 5 (2008): (1) method A, with a theoretical

background, can only be applied to shear walls with a tie-down at the

loaded leading stud in order to prevent uplift; and (2) method B

(together with the test protocol according to EN 594, 2008), which is

a soft conversion of the procedure developed in the United Kingdom

for racking strength given in BS 5268 (1996), (Porteous and Kermani,

2007), which can be used to design shear walls where the

corresponding stud is free to move vertically and the bottom rail is

anchored to the substrate. Method A corresponds to a fully anchored

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Introduction 7 shear wall, while method B corresponds to a partially anchored shear wall. Brittle failure of the bottom rail is not taken into account in Eurocode 5 (2008). Despite method B is used also for partially anchored shear walls, no recommendation is given on how to avoid possible bottom rail splitting. Vessby (2011) pointed out that both methods are to be considered as plastic methods, but if the bottom rail fails in a brittle manner, the applicability of plastic methods can be questioned. It is important to avoid brittle failure of the bottom rail in order to enable the development of the force distribution shown in Figure 1.4b and hence be able to apply plastic methods.

1.2 Aims and scope

The aim of this research is to identify the main factors influencing the splitting of the bottom rail in partially anchored timber frame shear walls. Further, the aim is to evaluate different developed models for calculating the splitting failure capacity of the bottom rail.

First, the splitting capacity of the bottom rail in partially anchored timber frame shear walls was measured in two experimental programs for single- and double-sided sheathing. Data was collected about the failure modes and failure loads of the bottom rail. Then theoretical models for the load-carrying capacity of the bottom rail, based on a fracture mechanics approach, were studied and validated. Two of the main parameters in the studied fracture mechanics models were the fracture energy and the tensile strength perpendicular to the grain values. Due to the orthotropic characteristics of wood, it was difficult to find values in literature for the same timber used in our studies and for the same crack orientation. It was then decided to carry out an additional matching experimental program, with bottom rail tests after which both fracture energy and tensile strength perpendicular to the grain were evaluated with specimens cut from the bottom rail specimen used in the tests. Then really explicit values were collected and used to compare model predictions to test results.

Specific questions addressed by the work presented in this thesis are:

¾ How do the varied parameters during the bottom rail tests, distance between the washer edge and the loaded edge of the bottom rail and the pith orientation of the bottom rail, influence the failure mode and load of bottom rail in partially anchored timber frame shear walls?

¾ Which of the evaluated models, based on a fracture mechanics

approach, show the best fit with the experimental results, in

terms of failure load, from the tests of bottom rail subjected to

uplift in partially anchored timber frame shear walls?

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8 Introduction 1.3 Limitations

The research has several limitations. All tests performed during the study were short-term tests. No full size shear wall has been tested;

however, data has been collected from previous studies. The cross section of the bottom rail used in the experiment was always the same:

120×45 mm. The species was spruce (Picea Abies). Only hardboard sheathing 8 mm from Masonite AB was used in the tests. During the bottom rail tests a small distance, 25 and 50 mm, between the nails in the sheathing-to-framing joints was used. This distance was applied, despite it is not a distance used in reality, in order to have a strong sheathing-to-framing joint and obtain splitting as the failure mode of the bottom rail. Finally, all models derived and validated in this study are 2D models that do not take into account that the anchor bolts were discretely placed along the bottom rail and are based on linear elastic fracture mechanics, even if a nonlinear approach is recommended for wood.

1.4 Outline of the thesis

This thesis is divided in two parts: part I gives a summary of the research carried out, while part II collects all journal articles written. In part I some additional information not included in part II are included;

a literature review on shear wall modelling.

Part I

This part is divided in eight chapters. Chapter 2 gives a literature review on shear wall modelling and the fracture mechanics concepts used in the thesis are included. Chapter 3 gives a background on the experimental studies. Chapter 4 collects the models evaluated for calculating the splitting failure capacity of the bottom rail. Chapter 5 collects the test results. Chapter 6 is a collection and discussion of the main findings of the study while chapter 7 summarizes the main conclusions. Finally chapter 8 gives suggestions about the future work.

Part II

Paper I “Splitting capacity of bottom rail in partially anchored timber frame shear walls with single-sided sheathing” by Giuseppe Caprolu, Ulf Arne Girhammar, Bo Källsner and Helena Lidelöw was published in The IES Journal Part A: Civil &

Structural Engineering in May 2014, 7:83 – 105. Giuseppe Caprolu’s

contribution to this paper was participating and performing the bottom

rail tests, evaluating the test results and carrying out the analysis

suggested by the senior authors. Furthermore, the introduction, the

experimental part of the paper including test results presentation, was

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Introduction 9 written by Caprolu. The experimental purpose was to study the influence of the distance between the edge of the washer and the loaded edge of the bottom rail and of the pith orientation on the failure mode and the failure load of the bottom rail with single-sided sheathing subjected to uplift in partially anchored timber frame shear walls.

Paper II “Splitting capacity of bottom rail in partially anchored timber frame shear walls with double-sided sheathing” by Giuseppe Caprolu, Ulf Arne Girhammar and Bo Källsner was published online in The IES Journal Part A: Civil &

Structural Engineering in November 2014. Giuseppe Caprolu’s contribution to this paper was participating and performing the bottom rail tests, evaluating the test results and carrying out the analysis.

Furthermore, the introduction, the experimental part of the paper including test results presentation, the analysis and the discussion were mainly written by Caprolu. The experimental purpose was to study the influence of the distance between the edge of the washer and the loaded edge of the bottom rail and of the pith orientation on the failure mode and on the failure load of the bottom rails with double- sided sheathing subjected to uplift in partially anchored timber frame shear walls.

Paper III “Analytical models for splitting capacity of bottom rails in partially anchored timber frame shear walls based on fracture mechanics” by Giuseppe Caprolu, Ulf Arne Girhammar and Bo Källsner was submitted to Engineering Structures in November 2014. Giuseppe Caprolu’s contribution to this paper was to provide the experimental background and performing the analysis.

Furthermore, the introduction, the experimental background of the paper, and part of the analysis and the discussion was written by Caprolu. The purpose was to present and validate analytical models based on a fracture mechanics approach, able to predict the splitting capacity of bottom rails.

Paper IV “Fracture mechanics models for brittle failure of bottom rails due to uplift in timber frame shear walls” by Jørgen L. Jensen, Giuseppe Caprolu and Ulf Arne Girhammar was submitted to Structural Engineering and Mechanics in November 2014.

Giuseppe Caprolu’s contribution to this paper was to provide the

experimental background. Furthermore, the experimental part of the

paper including test results presentation was written by Caprolu. The

purpose was to present and validate additional analytical models based

on a fracture mechanics approach, able to predict the splitting capacity

of bottom rails.

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10 Introduction

Paper V “Comparison of models and tests on bottom rails in timber frame shear walls experiencing uplift” by Giuseppe Caprolu, Ulf Arne Girhammar and Bo Källsner was submitted to Material and Structures in November 2014. Giuseppe Caprolu’s contribution to this paper was participating and performing the bottom rail, the fracture energy and the tensile strength perpendicular to the grain tests, evaluating the test results and carrying out the analysis.

Furthermore, the introduction, the experimental part of the paper

including test results presentation, the analysis and the discussion were

mainly written by Caprolu. The purpose was to have explicit values for

evaluating tests and analytical results in order to be able to state which

of the previously presented models show the best fit with the bottom

rail test results.

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Theoretical chapter 11

2 THEORETICAL CHAPTER

The chapter starts with a literature review of modelling of shear walls. Due to the high number of studies in this area, the most important studies are chosen on basis of the number of citations. Studies relative to seismic action are not taken into account. The review is grouped based on the modelling used for their derivation: (1) elastic; (2) plastic; and (3) finite element. Where possible, the derived equation for the load-carrying capacity of shear walls is presented. The two official design methods given in Eurocode 5 (2008) are explained. The purpose of the literature review was to highlight strong, weak and missing points of the different methods and also to show the simplicity of plastic methods, which, if the splitting of the bottom rail can be avoided, could be applied to partially anchored shear walls. Finally, a theoretical background for linear elastic fracture mechanics (LEFM), used to derivate the failure load models presented for the bottom rail, is explained.

The notation used in this chapter does not follow the general thesis notation.

2.1 Modelling of shear walls

Wood shear walls have been a research subject since the 1920’s with activities focused both on experimental and theoretical modelling approaches, Källsner and Girhammar (2009). Dolan and Foschi (1991) pointed out that the construction of timber buildings today is not the same as decades ago. Multifamily structures are larger. In addition, concrete overlayments on floors, concrete tile on roofs, and other new, heavier materials are used in the upper stories for fire protection, sound control, aesthetics, and reduced cost. Due to these and other changes to the construction of timber buildings, the assumption of past experience proving the reliability of timber structures is questionable.

Therefore, modelling of wood shear walls has evolved over the last

three decades from simple equations for the prediction of strength,

stiffness and deformation to complex nonlinear finite element models

detailed enough to include nonlinear elements for each fastener, van de

Lindt (2004). Models have been developed both for hand calculation

and computer based numerical models, usually based on the finite

element method. These are based both on linear elastic and nonlinear

elastic properties. They are applicable to fully and, in a few cases, to

partially anchored shear walls, both for shear walls with and without

openings and both for static and dynamic loads. Further, models have

been developed considering the influence of vertical loads and lateral

walls.

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12 Theoretical chapter

Some of the simplest, and most used, models for analysis of the capacity of shear walls are based on the theory of elasticity, Vessby (2011). The basic assumptions for the elastic models are: rigid framing and sheathing, framing members connected by frictionless hinges and bottom rail assumed to interact fully with the foundation. The results in these models are determined using the elastic approach where the shear wall capacity is based on the most loaded fastener. The plastic approach has the potential to specify more realistic load paths than is the case in an elastic analysis. The fasteners are assumed to reach their maximum capacity and, except for the corner ones, carry the full design load. The framing members are assumed to be completely flexible, which implies that the force distribution from the fasteners will become parallel with the framing members. This can be of great importance for load levels approaching the ultimate capacity as the load transferred by a single fastener changes both in terms of magnitude and direction. According to the assumptions above, plastic design methods are the only method that may be used to design partially anchored timber frame shear walls, on condition that a ductile behaviour of the sheathing-to-framing joint is provided and splitting of the bottom rail avoided.

Källsner et al. (2001) pointed out that the elastic model and the plastic lower bound model give almost the same load-carrying capacity for the shear wall. However the distribution of the fastener forces is fairly different, as shown in Figure 2.1.

a) b) c)

Figure 2.1 (a) A shear wall unit built up of a timber frame and a sheet; (b) forces

acting on the sheet according to a linear elastic model; and (c) forces acting on the sheet according to a plastic lower bound model.

Källsner et al. (2001) highlighted that at ultimate limit state the force distribution according to the plastic lower bound model can be justified for different reasons. One reason is that the joints between the timber members often tend to yield, which means that the force components perpendicular to the length direction of the timber members cannot be fully built up. Another reason is that at high loads some bending deformations in the timber members can almost always be seen, that also lead to reduced force components perpendicular to the timber members.

F F

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Theoretical chapter 13 2.1.1 Elastic models

In this section, where possible, equations derived in the presented studies are shown. The parameters in the equations refer to Figure 2.2.

Figure 2.2 Sheet dimensions and nail patterns.

Tuomi and McCutcheon (1978) developed a method based on the energy formulation where linear elastic nails absorb the internal energy and the external energy is given by the racking load. The model is able to predict the racking load of frame panels. The input data required are the panel geometry, the number and spacing of nails, and the lateral resistance of a single nail. Both small-scale and full-scale tests were run to verify the accuracy of the model. The theoretical results were found to give close agreement with experimental data for two panel sizes.

The total racking strength F of a panel is computed by:

n m

p

² na ² nb ² ma ² mb

f

F r K

ª ¬

K

w K

c K

w K

c K

º ¼ (2.1) where the subscripts p and f are the contribution of the perimeter and field nails, respectively, r is the individual nail resistance,

w H H_f

,

c B Bf

, according to Figure 2.2, and

K_n

,

K_m

,

K_na

,

K_nb

,

K_ma

, and

K_mb

are given in Tuomi and McCutcheon (1978). The model is elastic and

fully anchored to the foundation. McCutcheon (1985) highlighted that

this method assumes a linear load-displacement relationship for a single

nail, while in reality it is highly nonlinear, hence it cannot adequately

predict the real behaviour of the wall. Further, as pointed out by

Källsner and Girhammar (2009), the applicability of this model is

limited by the hypothesis chosen. In fact the corner fasteners are

supposed to displace along the diagonals of the sheathing, but this is

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14 Theoretical chapter

true only if the same number of fastener spacings are used in the rails and in the studs.

Itani et al. (1982) presented a methodology for calculating the racking performance of sheathed wood-stud walls with and without door and window openings. In their model each sheet is replaced by a pair of diagonal springs, with the stiffness of each spring calculated from the stiffness of an individual nail of the sheathing-to-framing joint. The stiffness of the diagonal springs K is calculated according to:

2 2

2 1 1

cos sin

4 3

k n m

K n m

n

E

m

E

ª § · º

« ¨ ¸ »

© ¹

¬ ¼ (2.2) where k is the nail slip modulus, n, m, and ȕ are given in Figure 2.2.

An equation was then used to fit the load F to the slip of the nails. The equation was based on unpublished experimental data collected at the U.S. Forest Products Laboratory. The model is based on Tuomi and McCutcheon (1978), hence the same objections about the linear behaviour of the fasteners may be made. Further, when calculating the stiffness, only the perimeter fasteners of the sheathing-to-framing joints are considered, while internal ones are neglected. However, this may be a good approximation since they contribute with only 5% of the total stiffness. The connection between bottom rail and foundation is modelled with linear springs. Finally, the model does not take into account the load carried by the parts above and below the openings. In Rainer et al. (2008), where three mechanics-based models were compared, it was found an increase of calculated load-carrying capacity between 20% and 34% when the panels above and below the openings were added to the prediction.

Easley et al. (1982) derived equations for the sheathing fastener

forces, for the linear shear stiffness of a wall and for the nonlinear shear

load-strain behaviour of a wall. The equations were based on

deformation patterns observed during testing, which were verified

using linear and nonlinear finite element analyses. For the 2D finite

element model the sheets were modelled as plane stress isotropic

element with eight nodes. The frame was also modelled with eight-

nodes and a linear isotropic material. Two springs were used to model

the sheathing-to-framing connection, one in each perpendicular

direction. They concluded that their equation for sheathing forces

should only be applied in the linear range, with the exception of the

side and maximum end fastener force, which is accurate well into the

nonlinear range. The model is elastic and fully anchored to the

foundation. Further, as pointed out by Källsner and Girhammar (2009),

the fasteners along the vertical studs were assumed to be loaded only in

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Theoretical chapter 15 the vertical direction. This assumption is an approximation. Due to the number of equations, they are not presented here.

Gupta and Kuo (1985) presented a simple numerical model, based on a generalized coordinate approach to derive equilibrium equations, to represent the shear behaviour of shear walls. Nonlinear properties were used for the sheathing-to-framing joints. The model includes the bending stiffness of the stud and shear stiffness of the sheathing. The model was compared to a finite element model and shear wall tests performed by Easley et al. (1982) and Foschi (1982). The comparison showed the adequacy of the model. They concluded that their model was accurate and simple enough to be used in repetitive analysis, e.g.

nonlinear dynamic analysis. This model considers only fully anchored shear walls. Further, Robertson (1980) indicated that the shear strength per unit length of wall increases with the increase in the vertical loading on the wall and with the increase in the length of the wall.

This dependence of the shear strength on the vertical load and the wall length cannot be explained by the model of Gupta and Kuo (1985). At a later stage, Gupta and Kuo (1987a) made a modification of the model, taking into account the uplift of the studs (assuming fully anchored bottom rail). The proposed model had five degrees of freedom (DOF) for a single-storey wall and two additional DOF for walls of two or more stories. The studs were modelled as continuous through all stories and each storey had a separate sheet. In Gupta and Kuo (1987b) an analytical three-dimensional model of a complete house was presented. A major part of the effort went into perfecting the shear wall behaviour suitable for house analysis. The model was extended in order to consider uplift of the bottom rail, in addition to that of the stud. The model prediction was compared to the test results from a full scale house, presented in Tuomi and McCutcheon (1974), giving results that were in good agreement with the experimental results. The derived equations are given in matrix form and hence they are not presented here.

Mallory and McCutcheon (1987) extended a previous elastic model for shear wall performance developed by McCutcheon (1985), to model the nonlinear racking load-displacement behaviour of fully anchored shear walls sheathed on both sides with dissimilar materials.

Four types of curves were used to model the fastener load slip: power,

logarithmic, hyperbolic and asymptotic, with the latter found to give

the best agreement with test results. The model prediction was

compared to the results of numerous small wall tests, and predicted the

racking behaviour well. The derived equation for the racking load F

was:

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16 Theoretical chapter

2 f

f

F S Q

Z Q

§ ' ·

¨ ' ¸

© ¹

¦ (2.3) where Q is the wall horizontal racking displacement, ȴ

f

is the fastener slip and S and Z are constants.

Schmidt and Moody (1989) developed a simple structural analysis model to predict the nonlinear deformations of three-dimensional light frame buildings under lateral load. Openings are not included in the model. The model is based on the energy method and is an extension of the previous work of Tuomi and McCutcheon (1978), which is combined with nonlinear load-slip curves for fasteners presented by Foschi (1977) and McCutcheon (1985). A comparison of the predicted behaviour to the results from two full-scale house tests, Tuomi and McCutcheon (1974) and Boughton and Reardon (1984), reveals reasonable agreement with the test results. The derived equations are given in matrix form and they are not given here.

Filiatrault (1990) developed a simple structural analysis model to predict the behaviour of timber shear walls under lateral static loads and earthquake excitations. The model is restricted to two-dimensional shear walls with arbitrary geometry of the framing, sheathing and connections, and wall discontinuities, i.e. openings. Nonlinear load-slip characteristics of the fasteners are used in a displacement-based energy formulation to develop the static and dynamic equilibrium equations.

The model was verified with full-scale shear wall racking and shaketable tests, and was found to be accurate. The derived equations are given in matrix form not given here.

Källsner and Girhammar (2009) presented an analysis of fully anchored light-frame timber shear walls. The analysis was based on an elastic model with the assumption of a linear elastic load-slip relation for the sheathing-to-framing joints. Only static loads were considered.

Equations both for the load-carrying capacity and the deformation of the shear walls for ultimate and serviceability limit state were derived.

Openings in the shear walls were not considered. Forces and

displacements of the fasteners and sheathing were also derived. Other

influences discussed were: discrete point or continuous flow per unit

length modelling of the fasteners, effect of different patterns and

spacing of the fasteners, influence of flexible framing member and shear

deformations in the sheets and also the effect of vertical loads. The

model was compared to the results from an experimental study and

reasonable agreement was found. The equation proposed for the

horizontal load-carrying capacity F of the wall unit was:

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Theoretical chapter 17

2 2

1 1

ˆ ˆ

corner corner

n n

i i

F r

x y

H x y

ª º ª º

« » « »

¬ ¦ ¼ ¬ ¦ ¼

(2.4)

where r is the shear capacity of the fastener, H as given in Figure 2.2 and

ˆx

and ˆy are the fastener coordinates referring to the new coordinate axes, which are referred to the centre of gravity of the fasteners.

2.1.2 Finite element models

Foschi (1977) presented a structural analysis for wood diaphragms based on finite element model. Four different structural elements were considered in the analysis: the sheet, assumed to be elastic and orthotropic, the frame, represented by linear beam elements and the connections between frame members and sheet-frame connections, assuming a nonlinear behaviour. A comparison was made with experimental results on 6×18 m plywood and decking roof diaphragms. The comparison showed that the analysis gives reliable estimates for diaphragm deformations and is capable of providing an approximation for ultimate loads based on connection yielding.

Falk and Itani (1989) presented a two-dimensional finite element model for analysing the nonlinear load displacement of vertical and horizontal wood diaphragms. Their formulation included a nonlinear finite element model that accounted for the distribution and stiffness of fasteners connecting the sheet to the framing. A parametric study was performed and it showed that both nail stiffness and nail spacing, the latter with a greater effect, influenced the diaphragm stiffness. Blocking was shown to increase the diaphragm stiffness due to the greater number of nails used with blocking and the increased frame action provided. A comparison of the model results with experimental tests reported in Falk and Itani (1987) indicated a good prediction. This model is a respond to the finite element model proposed by Itani and Cheung (1984) for the static analysis of wood diaphragms. That model needed a large number of DOF when modelling large diaphragms. The model presented by Falk and Itani (1989) require fewer DOF and gives a better representation of the distributed fasteners if larger ceiling and floor diaphragms have to be analysed.

A numerical model, based on a finite element analysis procedure for

nonlinear static analysis of wood shear walls was developed by Dolan

and Foschi (1991). The model is an improved version of that

developed by Foschi (1977) and the improvements are the possibility

to include: (1) nonlinearities in the sheathing due to bending and

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18 Theoretical chapter

buckling of the sheathing; (2) modification of the fastener in the sheathing-to-framing joint in order to include three directions of movement and the ultimate capacity of the connector; and (3) the bearing between adjacent sheathing elements. The model has been verified by comparison with the load-deflection curves from full-scale shear wall tests presented by Dolan (1989) and good prediction was found. The authors conclude that their model is general and capable of modelling irregular shapes as well as adhesive connections.

In order to reduce the total number of DOF Kasal and Leichti (1992) developed a two-dimensional model that was equivalent to a detailed three-dimensional model. The equivalent model was formulated using equivalent energy concepts, and yielded the global behaviour of the structure in reasonable time. The model can treat a wall with or without openings.

2.1.3 Plastic models

In Ni and Karacabeyli (2000; 2002) one mechanical-based method and one empirical method were developed to account for effects of vertical loads and perpendicular walls on the performance of shear walls with and without hold downs. The methods were found to be in reasonable agreement with test data from a previous study. The proposed equation to calculate the lateral capacity F for the mechanical-based method is given as:

²

p

1 2

F f L

IJ J J (2.5) Where f

_p

is the plastic capacity of a panel per unit length, I

P f H_R _p

, where P

_R

is the uplift restraint force on the end stud of a shear wall segment H is given in Figure 2.2, and J H L , where L is the full wall length. The mechanical-based option has been adopted in Canada in the CSA-O86 (2001) Standard for wood design and in the Wood Design Manual. Eq. (2.5) was then changed by simply introducing the hold down effect reduction factor J

_hd

, to Eq. (2.5). The reduction factor is calculated as:

2 ij

hd

1 2

P H H

1.0

J V B B

§ ·

_ij

and V

_hd

as given in Ni and Karacabeyli (2002) and H and B as given in Figure 2.1.

If the shear wall is fully anchored J

_hd

is considered as being unity,

otherwise it is determined using Eq. (2.6). The model is not able to

take into account openings in the shear wall.

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Theoretical chapter 19 Later, Källsner et al. (2001; 2002), developed a plastic lower bound method, meaning that the force distribution was chosen in order to fulfil the conditions of force and moment equilibrium. The method is able to calculate the load-carrying capacity of fully and partially anchored timber frame shear walls at ultimate limit state. The model covers only static loads and can only be applied when mechanical fasteners with plastic characteristics are used. The influence of vertical loads is also taken into account. Only walls without openings were dealt with in this study. Many equations have been presented depending on the anchorage system and external loads acting on the shear walls. Due to the number of equations, they are not presented here.

In Källsner and Girhammar (2004) a plastic lower bound method was presented to study the influence of the stud-to-rail joint on the load-carrying capacity of partially anchored timber frame shear walls.

The calculations showed that considering this, the load-carrying capacity can be increased by 10 to 15%. In this method the full vertical shear capacity of the wall was utilized, which not fully fulfil the conditions of equilibrium. The calculated load-carrying capacity was equal or slightly higher than the method presented in Källsner et al.

(2001; 2002), but it was much easier to calculate. In Källsner and Girhammar (2005) this theory was presented in a simple format and it was shown that the theory can also be applied to shear walls with openings. Vertical point and distributed loads acting on the wall were considered. The model assumptions were:

x The model covers only static loads;

x The sheathing-to-framing joints in the vertical studs and top rail are assumed to transfer only shear forces parallel to the timber members;

x The sheathing-to-framing joints in the bottom rail are assumed to transfer forces both parallel and perpendicular to the bottom rail;

x The framing joints can transfer tensile or shear forces;

x Compressive forces can be transferred via contact between adjacent sheets and in the framing joints.

This analytical model has different advantages. It is able to calculate

the load-carrying capacity of shear walls with and without openings. It

can be used in design of shear walls with different sheet materials,

sheathing-to-framing joints, geometric layout, anchoring conditions

and load configurations. The main problem has been that the shear

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20 Theoretical chapter

walls are fastened to the substrate in different ways in different countries. This fact must be reflected in national codes but it is not.

The model can be applied to shear walls that are fully or partially anchored to the substrate, giving a solution to this problem. The authors derived easy used closed form equations for the wall configurations needed by a designer. They are presented in Källsner and Girhammar (2005).

2.1.4 Design method according to Eurocode 5

In Eurocode 5 (2008) two methods are given for the design of shear walls, method A and method B. Method A is based on a theoretical background, while method B is a soft conversion of the procedure developed in the United Kingdom for racking strength and given in BS 5268 (1996), (Porteous and Kermani, 2007). In both models, the design racking load-carrying capacity is based on the lateral design capacity of the individual fasteners in the sheathing-to-framing joints.

The capacity of the single fastener is then multiplied by the number of spacings between these connections and the design load-carrying capacity of the wall panel is obtained. If the wall assembly is composed of several wall panels, the total load-carrying capacity is given by the sum of their single load-carrying capacities. In method A, the capacity of areas around door and window openings in the wall panels are not considered to contribute to the total load-carrying capacity, while in method B no mention is made concerning this. The fastener spacing is constant along the shear wall perimeter and all fasteners are considered to reach their maximum lateral load capacity. It should be noted that the two methods have different boundary conditions: method A corresponds to a fully anchored shear wall while method B corresponds to a partially anchored shear wall, meaning that the studs are allowed to separate from the bottom rail when subjected to uplift and that the bottom rail can be subjected to transverse bending. It is obvious that the two methods are not consistent with one other, except in the case where vertical loads of sufficient magnitude to stabilize the wall are applied in method B. As already highlighted, the structural behaviour in the case of partially anchored shear wall introduces different failure modes than the fully anchored, for example splitting failure of the bottom rail could happen. However, no recommendations are given with respect to this.

Vessby (2011) pointed out that both methods are to be considered as methods based on theory of plasticity since they assume the same load (magnitude) being transferred by all the fasteners. However, no recommendations are given on any larger scale than parts of walls, i.e.

single shear walls. Much of the benefits of an overall plastic analysis,

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Theoretical chapter 21 with possibilities to e.g. include the effects of lateral walls, are not indicated and thus not regulated in Eurocode 5 (2008).

2.2 Fracture mechanics

Aicher et al. (2002) pointed out that when a body made of a solid material is loaded it will ultimately respond by undergoing large deformations or fracture. Fracture is the loss of contact between parts of the body resulting in a creation of two new surfaces and it is the topic of interest in fracture mechanics. The concern is partly with the microscopic mechanism, which govern the separation and partly with predictions from a macroscopic point of view. Of prime concern is the development of criteria and methods by which it is possible to predict the load-carrying capacity of structural members based on knowledge about the material properties. The factors that govern fracture are:

loading conditions, material properties, size and shape of body and defect in material or body.

Fracture mechanics is a branch of mechanics of materials. It is used in situations where large stress or strain concentrations arise, such as close to holes or notches. Serrano and Gustafsson (2006) highlighted that the geometrical features of timber structures, including the ultra- structure of the wood material, are such that flaws, cracks or sharp corners always induce stress or strain singularities. The presence of knots, drying cracks and other anomalies in timber also represent stress or strain concentrations, which can be considered as crack equivalents.

Consequently, traditional approaches based on stress and strain criteria, can give poor predictions of the load-carrying capacity in many cases, and a fracture mechanics approach can give better predictions.

Three basic types of loading and fracture are defined for a body, as

shown in Figure 2.3. Mode I is the opening mode, mode II is the in-

plane shear mode and mode III the out-of-plane shear mode. Usually

the bodies are not loaded in only one mode but in a combination of

them, giving a mixed mode loading.

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22 Theoretical chapter

Figure 2.3 The three modes of loading and fracture.

Since wood is an orthotropic material, the three directions (longitudinal, radial and tangential) give six possible orientations of the crack, as shown in Figure 2.4. The possible orientations are: RL, TL, LT, RT, LR and TR. In this notation the first letter indicates the direction normal to the crack plane while the second letter indicates the direction of the crack growth.

Figure 2.4 Crack orientations in wood.

Since there are three loading modes for crack orientation, there are a total of 18 crack situations, with different values of the crack resistance.

Fracture mechanics theory is divided in two branches, linear elastic fracture mechanics (LEFM) and nonlinear fracture mechanics (NLFM).

In LEFM the material under consideration is assumed to exhibit

linearly or very nearly linearly elastic behaviour right up to the point

where fracture occurs and it is supposed that all the available strain

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Theoretical chapter 23 energy goes into propagating a crack. However, in almost all materials there are several microstructural mechanisms that are capable of dissipating energy strain energy. If these microstructural mechanisms are taken into account NLFM should be used. The influence of the microstructural mechanisms depend on the size of the body compared to the fracture process zone. In Smith and Vasic (2003) a work aimed at identifying the crack evolution in softwood, including any restraining mechanisms due to bridging and micro-cracking at crack tips was reported. It was shown that behind the crack tip partially delaminated longitudinally oriented cells (tracheids) bridged the crack.

This fibre bridging provided crack closure forces proportional to the local crack opening displacement. Bridging was found to be the main mechanism of crack tip shielding in spruce and presumably other softwood species. It has an influence on the fracture energy.

Obviously, there is no contribution to the fracture energy from the bridging stresses prior to the crack initiation and evolution, but once established, the bridging zone was found to contribute with about 10%

to the total fracture energy release rate. When activated, they decrease the strain energy G in the sense that not all strain energy is used for the crack growth but a part of it will be dissipated by the microstructural mechanism and will increase the crack resistance R in the sense that the fibres will tend to counteract crack opening. Since the size of the fracture process zone is essentially invariant, its influence changes with the size of the specimen. Larger specimens will have behaviour closer to LEFM and smaller specimens closer to NLFM.

2.2.1 Strain energy release rate

The main point in fracture mechanics is to find a criterion that can be used in order to predict when a crack present in a body starts to propagate. Griffith (1921) made a study in order to provide a quantitative criterion for crack growth. His approach was to consider the thermodynamic equilibrium of a system with a crack.

The total energy 3 of a loaded system can be written as:

U L W

3

(2.7) where U is the elastic strain energy stored in the body loaded by an external force, L is the negative work of load due to the change in the potential energy of the system and W is the surface energy associated with the crack formation.

The Griffith criterion for crack growth can then be written as:

d d

L U W

A

(2.8)

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24 Theoretical chapter

where dA is the incremental change in the crack area. The left-hand side of Eq. (2.8) is commonly referred to as the strain energy release rate G, while the right-hand side of Eq. (2.8) is commonly referred to as the crack resistance R.

Therefore G is interpreted as the energy available to grow a crack of unit area, while R is interpreted as the energy required for propagating a crack of unit area. Hence G = R is considered as the critical condition for the crack propagation. The critical strain energy, G

c

, is the value of G when the crack starts to propagate and it is often used as a condition for crack growth. The value of G depends on the mode of loading, but in case of mixed mode fracture G = G

_I

+ G

_II

+ G

_III

, where the subscript refers to the mode of loading. This quantity can be measured by test of an elastic body subjected to a load, and it is given by the area under the load-displacement curve. The load can be either an applied load or a result of displacement control, but since it is important to have stable crack growth, displacement control is suggested. Serrano and Gustafsson (2006) pointed out that in order to avoid catastrophic failure upon and after reaching the maximum stress, the amount of strain energy released during the course of fracture must be less than or equal to the amount of energy needed to continue the fracture softening process. Smith et al. (2003) showed that a stable crack growth is possible only if displacement control is applied.

If the compliance C is introduced and defined as the reciprocal of the slope of the load-displacement curve, it has been shown that G becomes:

1

2

d

2 d

G P C

b a

(2.9) for both load and displacement control. In Eq. (2.9) b is the thickness of the specimen, P is the value of the force which cause the crack growth and a is the crack length. The failure load can then be obtained as:

^c

2 d

d

P C A

A

(2.10)

where A is the area of the crack considered. The area can be calculated using the initial crack length a

_c

according to Eq. (2.11) given in Serrano and Gustafsson (2006):

c 2

t

E c

a

S

f

(2.11)

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Theoretical chapter 25

where E is the modulus of elasticity and f

_t

is the tensile strength.

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Experimental studies 27

3 EXPERIMENTAL STUDIES

Three different types of experiments were carried out:

1. Splitting capacity of bottom rail in partially anchored timber frame shear walls with single- and double-sided sheathing;

2. Fracture energy of spruce (Picea Abies) in the RT and TR plane;

3. Tensile strength perpendicular to the grain of spruce (Picea Abies) in tangential and radial direction.

3.1 Splitting capacity of bottom rail 3.1.1 Material properties

The details of the test specimens were as follows:

x Bottom rail: spruce (Picea Abies), C24 according to EN 338 (2009), 45×120 mm;

x Sheathing: hardboard, 8 mm (wet process fibre board, HB.HLA2, EN 622-2 (2004), Masonite AB);

x Sheathing-to-framing joints: annular ringed shank nails, 50×2.1 mm (Duofast, Nordisk Kartro AB). The joints were nailed manually and the holes were pre-drilled in the sheets, Ø 1.7 mm;