EFFICIENT LOAD DISTRIBUTION ANALYSIS AND

STRENGTH PREDICTION OF BOLTED COMPOSITE

JOINTS AT VARIOUS LOADING RATES

Philip Anthony Sharos

Thesis submitted in fulfilment of the requirements for the Degree of Doctor of Philosophy at the Faculty of Science and Engineering,

University of Limerick

Candidate Supervisor: Dr. Conor T. McCarthy

Submitted to the University of Limerick: May 2016

i

ABSTRACT

Mechanical fasteners are used extensively in composite aircraft structures, offering a

cost-effective, reliable and repeatable joining method while also facilitating

disassembly for maintenance and repair. However, current design practices rely

extensively on experimentation and detailed numerical analyses which are time

consuming and expensive and thus significantly increase development and production

costs. Although three-dimensional finite element analysis can be used in-lieu of more

expensive experimental tests, current state-of-the-art computing technology cannot

facilitate analysis of large-scale sub-structures (e.g. panels, wing spars, etc.) with

potentially thousands of fasteners. The work conducted herein addresses this issue

through the development of highly efficient analysis methodologies for bolted

composite joints loaded at various rates.

Several highly efficient numerical methods are developed that can account for

laminate-laminate friction, bolt-hole clearance, non-linear material behaviour, bolt

failure and loading-rate effects. Validated against quasi-static and high-rate

experimental data, these methods have significantly advanced the state-of-the-art, due

to their ability to accurately predict the response of multi-row, multi-column joints in

seconds. Such an increase in efficiency is owed to the implementation of a series of

parameterised functions which model the local mechanical behaviour of the fastener

and fastened material. This approach allows the statistical variations arising from

manufacturing and material processing to be easily accounted for, a feat that is

impractical using traditional modelling techniques and near-impossible

experimentally. Furthermore, these methods have been developed into user-friendly,

stand-alone joint analysis tools and a user-defined finite element which is

straightforward to implement in commercial finite element software.

The developed analysis methods were used to investigate the effects of loading rate

and relative spacing between fasteners in multi-bolt joints. Although loading rate

effects primarily manifest as changes in joint failure loads and energy absorption, the

propagation of stress waves also leads to variations in load distribution of up to 1.7%

in the joints considered. Furthermore, it was found that a strain shielded region exists

in all multi-fastener arrays and consequently any bolts positioned within this region

carry a reduced proportion of load. When load is evenly distributed across the joint,

increased failure initiation loads are to be expected, however, the most effective

method of augmenting joint strength is through the use of additional fasteners.

Additionally, the use of a load-path visualisation methodology gave novel insight into

the role of fibre orientation on load transfer in bolted multi-directional laminates. It

was found that load is transferred in a trajectory that is strongly dependent on fibre

orientation, which is important in the distribution of load in multi-fastener joints.

iii

DECLARATION

The substance of this thesis is the original work of the author and due reference and

acknowledgement has been made, where necessary, to the work of others. No part of

this thesis has already been submitted and is being concurrently submitted in

candidature for any other degree.

Candidate Supervisor

Philip A. Sharos Dr. Conor McCarthy

Signature Signature

Date Date

Examination Board

Chairman: Dr. Jeremy Robinson

External Examiner: Prof. Carlos Santiuste

Internal Examiner: Prof. Michael McCarthy

v

ACKNOWLEDGEMENTS

First and foremost, I would like to express my utmost gratitude to my supervisor,

Dr. Conor McCarthy, for his guidance, encouragement and patience throughout my

studies in UL. His knowledge in finite element modelling and in the mechanics of

composite bolted joints was invaluable. I am grateful to have had the opportunity to

work under his supervision.

I would like to gratefully acknowledge the financial support provided to be by the Irish

Research Council (IRC) and the Donegal County Council.

I would also like to thank Dr. Brian Egan and Dr. Paddy Gray for making their

experimental data available to me (work that was frequently referenced throughout

this thesis). Without access to this data, much of my work would not have been

possible. Also sincere thank you to Dr. Ronan O’Higgins who kindly answered my

numerous questions.

A special thanks to the friends I have made during my time in UL and those who have

helped me along the way including (in no particular order!) Brian, Donal, Rob, Ned,

Brian, Rory, Niall, Jon, John, Kev, James and my good friends in the Skydive Club.

A quick shout out to Mark, I’m sorry Dan and I undoubtedly ruined your beloved

Counter-Strike for you! And to Dan, with whom I’ve shared too many Chinese

takeaways with during all those late nights in the office. Also, I would be remiss if I

did not mention my good friend Chrispy for keeping me relatively sane with the

occasional Friday night pint(s).

Finally I would like to thank my family for their unconditional love and support, not

just during this process but throughout my life.

vii

CONTENTS

List of Figures xi

List of Tables xxi

Nomenclature xxiii

Chapter 1: Introduction

1.1 Background and Motivation 2

1.2 Objectives 6

1.3 Thesis Outline 7

Chapter 2: Literature Review

2.1 Introduction 10

2.2 Mechanical Behaviour of Bolted Joints 10

2.2.1 Terminology 10

2.2.2 Failure Modes of Bolted Joints 12

2.2.3 Material and Lay-up 15

2.2.4 Geometry 16

2.2.5 Fastener Type 17

2.2.6 Bolt-hole Clearance 18

2.2.7 Bolt Torque and Lateral Constraint 20

2.2.8 Loading Rate 24

2.2.9 Multiple Fasteners 29

2.3 Detailed Finite Element Modelling of Bolted Joints 33

2.3.1 Simulation of Elastic Joint Behaviour 33

2.3.2 Simulation of Joint Failure 35

2.4 Efficient Analysis Methods 39

2.4.1 Closed-Form Approaches 39

2.4.2 Finite Element Analysis 43

2.5 Discussion of Literature Review 49

viii

2.5.1 Mechanical Response of Bolted Joints 49

2.5.2 Detailed Finite Element Analysis 51

2.5.3 Efficient Analysis Techniques 51

Chapter 3: Analytical Model for Strength Prediction of Joints Under

Quasi-Static and High-Speed Loading

3.1 Introduction 54

3.2 Problem Description 54

3.3 Model Development 55

3.3.1 Undamaged Joint Response 58

3.3.2 Damaged Joint Response and Rate Effects 61

3.4 Model Validation 68

3.4.1 Single Fastener Results 70

3.4.2 Multi-Fastener Results 73

3.5 Conclusions 76

Chapter 4: Development and Validation of Finite Element Code for

Multi-Fastener Joints

4.1 Introduction 78

4.2 Computational Efficiency of FE Solution Methods 78

4.3 Mesh Generation and Element Stiffness Matrices 82

4.4 Direct Solution Method 84

4.4.1 Fastener Element Behaviour 84

4.4.2 Boundary Conditions and Loading 86

4.5 Virtual Fastener Method 87

4.5.1 Fastener Reaction Forces 88

4.5.2 Displacement Field Approximation 89

4.5.3 Summary of Virtual Fastener Method 93

4.6 Model Validation 94

4.6.1 Statistical Variation of Fastener Properties 94

ix

4.6.2 Effect of Missing Fasteners 101

4.6.3 Computational Efficiency 103

4.7 Graphical User Interface 106

4.8 Conclusions 109

Chapter 5: User-Defined Finite Element for Bolted Joints in Explicit

FE Solvers

5.1 Introduction 112

5.2 Model Development 114

5.2.1 Overview 114

5.2.2 User Element Behaviour 116

5.2.3 Composite Material Modelling 124

5.3 Model Validation 126

5.3.1 Single Bolt Results 126

5.3.2 Multi-fastener Quasi-static Results 129

5.3.3 Dynamic Results 135

5.4 Effect of velocity on load-distribution 137

5.5 Conclusions 143

Chapter 6: Analysis of Load Distribution in Multi-Fastener Joints

6.1 Introduction 146

6.2 Load Path Study 146

6.2.1 Theoretical Overview 146

6.2.2 Effect of Fibre Orientation 149

6.2.3 Missing Fasteners 156

6.3 Fastener Positioning 161

6.4 Conclusions 167

Chapter 7: Conclusions

7.1 Introduction 170

x

7.2 Notes on the Mechanical Response of Bolted Joints 170

7.2.1 Variations in Individual Bolt Response in Multi-Fastener Joints 170

7.2.2 Effects of High-Rate Loading 171

7.2.3 Effect of Fastener Position 172

7.2.4 Effects of Material and Layup 172

7.2.5 Guidelines for Design 173

7.3 Highly Efficient Numerical Methods 174

7.4 Recommendations for Future Work 177

Bibliography 179

Appendix A: Approximation of Computational Time Required for

Finite Element Solver Methodologies

A.1 Overview A-1

A.2 Approximation for Common Matrix Operations A-1

A.2.1 Matrix Multiplication A-2

A.2.2 Matrix Inversion (Gauss-Jordan Elimination) A-2

A.2.3 Gauss-Elimination with Back Substitution A-4

A.3 Number of Scalar Operations for FE Solution Methods A-5

A.3.1 Direct Solution Method A-5

A.3.2 Virtual Fastener Method (VFM) A-5

A.3.3 Explicit Method A-6

A.4 Notes on the Efficiency of the VUEL Model A-7

Appendix B: Input File Configuration and VUEL Subroutine for User-

Defined Finite Element

B.1 Introduction A-9

B.2 ABAQUS Input File Configuration A-9

B.3 MATLAB Script to Generate UEL Properties A-11

B.4 VUEL FORTRAN Subroutine A-25

xi

LIST OF FIGURES

Chapter 1

Figure 1.1 Boeing 787 showing breakdown of materials used (redrawn from

Georgiadis et al. 2008)).

2

Figure 1.2 Examples of materials and process development acceleration

using computational tools under the AIM-C project (a)

traditional testing supported analysis approach (b) analysis

approach supported by experience, testing and demonstration

(redrawn from National Research Council (2004)).

4

Figure 1.3 Certification procedure for composite materials (redrawn from

MAAXIMUS (2009)).

5

Chapter 2

Figure 2.1 Definition of geometric parameters of bolted joints (redrawn

from McCarthy (2003)).

11

Figure 2.2 Failure modes of mechanically fastened joints (redrawn from

Niu (1992)).

13

Figure 2.3 Definition of (a) clearance and (b) contact angle. 19

Figure 2.4 Effect of bolt torque on initial load-displacement curve of

single-bolt, single-lap joint with clearance (redrawn from

McCarthy et al. (2002)).

21

Figure 2.5 (a) Loads carried by various interactions in single-lap joint (b)

Bolt and total friction load distribution in multi-bolt joint with

variable hole clearance, C1=neat-fit, C4=240 µm clearance

(redrawn from McCarthy et al. (2005)).

23

Figure 2.6 (a) Load-displacement behaviour of pinned and clamped

laminated (redrawn from Kelly & Hallström (2004)); (b) Effect

of clamping pressure on ultimate bearing strength of bolted

joints (redrawn from Park (2001)).

24

xii

Figure 2.7 Quasi-static and high rate stress strain curves for: (a) Transverse

compression and; (b) shear in unidirectional carbon/epoxy

composites (redrawn from Hsiao & Daniel (1998)).

26

Figure 2.8 Comparison of quasi-static and dynamic bearing response of

composite joints redrawn from (a) Ger at al. (1996); (b) Pearce et

al. (2009).

28

Figure 2.9 Bearing-bypass loading within a multi-fastener joint (redrawn

from Crews & Naik (1987)).

30

Figure 2.10 Determining bypass loads in a multi-bolt joint (redrawn from

McCarthy et al. (2005)).

31

Figure 2.11 Loading of single lap joint with (a) no deformation; (b)

deformation with rigid members; (c) deformation with elastic

members (redrawn from Eriksson et al. (1995)).

32

Figure 2.12 Bolt load distribution in 3-bolt double-lap composite joint with

(a) equal clearance at each hole (neat-fit); (b) 240µm clearance at

hole 1, neat-fit at holes 2 & 3 (redrawn from McCarthy et al.

(2005)).

33

Figure 2.13 Flowchart of the progressive damage model (redrawn from

Tserpes et al. (2001)).

36

Figure 2.14 (a) Meshing strategy and material property assignments in the

bolt-hole region and (b) cohesive element locations (redrawn

from Egan et al. (2015)).

39

Figure 2.15 Figure 2.15 Spring element model for a three-bolt joint (redrawn

from McCarthy et al. (2006)).

41

Figure 2.16 Geometry of bolted hybrid joint (redrawn from Caccese et al.

(2007)).

45

Figure 2.17 Video sequences from bird strike on specimen versus simulation

results with rivet failure law: (a) t = 4.5ms; (b) t = 10ms

(redrawn from McCarthy et al. (2004b)).

46

xiii

Figure 2.18 2.18 PLINK model load-displacement behaviour (redrawn from

Gunnion et al. (2006)).

47

Chapter 3

Figure 3.1 Typical load-displacement curve for single-bolt, single-lap joint:

(a) complete load displacement response; (b) enlarged view of

undamaged loading behaviour.

55

Figure 3.2 Three-bolt, single-lap joint: (a) illustration; (b) corresponding

spring/mass system (redrawn from McCarthy et al. (2006).

56

Figure 3.3 Contact area for calculating Ks in highly torqued joint. 58

Figure 3.4 Derivation of spring stiffness term for tapered laminate section:

(a) Equivalent spring model for single bolt joint; (b) Enlarged

view of taper section.

60

Figure 3.5 Possible methods of inserting plies to taper up laminate and

definition of ply drop off ratio in relation to laminate geometry.

61

Figure 3.6 Experimental results of a single-bolt joint tested quasi-statically

and at loading rates of 5 m/s and 10 m/s, (Redrawn from Egan

et al. (2013))

62

Figure 3.7 Illustration of critical damage variables used in analytical model. 63

Figure 3.8 Comparison between conic and cubic damage approximation

functions for a single bolt joint: (a) Low energy case; (b) high-

energy case.

64

Figure 3.9 Use of conic curves in the prediction of the joint non-linear

stress strain behaviour: (a) elliptical curve and (b) enlarged view

of hyperbolic curve.

67

Figure 3.10 (a) Single-bolt geometry, and (b) multi-bolt geometry. See Table

3.1 for values of e, w, p, L and g.

68

Figure 3.11 Comparison of experimental and model load-displacement

curves for single-bolt (SB) joints (experiments redrawn from

Egan et al. (2013) and Gray et al. (2014a), QS = quasi-static, A, C

71

xiv

and E correspond to the layups used in joint, FF=fastener

failure, FP=fastener pull-through).

Figure 3.12 Comparison of experimental and model load-displacement

curves for single-both E-laminate joints accounting for residual

fastener strength: (a) 5m/s, and (b) 10m/s.

72

Figure 3.13 (a) Fastener numbering in single-column joint. (b) & (c) Quasi-

static load displacement curve of C-layup and E-layup joints

with equal fastener properties, respectively. (d) & (e) Individual

fastener load displacement curves in C-layup and E-layup joints

with equal fastener properties, respectively. (f) & (g) Individual

fastener load displacement curves in C-layup and E-layup joints,

respectively, with different values of energy absorption assigned

to each fastener (values given in Table 3.5).

74

Figure 3.14 Comparison of experimental and model load-displacement

curves for multi-bolt joints (experiments redrawn from Egan et

al. (2013) and Gray et al. (2014b), QS = quasi-static, A, C and E

correspond to the layups used in joint).

75

Chapter 4

Figure 4.1 Comparison of normalised CPU times and number of

mathematical operations required for basic matrix operations

(GEBS – gauss elimination with back substitution).

80

Figure 4.2 Comparison of theoretical computational times for VFM and

direct solution method (a) times normalized w.r.t the direct

method; (b) Ratio between direct and VFM CPU times.

81

Figure 4.3 Disadvantages of paving method (a) interference of opposing

elements (b) element size difference between opposing fronts:

(redrawn from Lee et al. (2003)).

83

Figure 4.4 Mesh post-processing operation to account for specific fastener

locations.

84

xv

Figure 4.5 (a) Integration of fastener element with composite laminate

mesh; (b) Deflection of fastener element due to bearing loads.

85

Figure 4.6 Boundary conditions used and example finite element mesh for

the direct solution method.

86

Figure 4.7 Loads and boundary conditions used for Virtual Fastener

Method (VFM). (Note that XJ symbolises the rigid body motion

applied to Laminate 2 and is not used as a boundary condition

in the solution of Eq. (4.6)).

87

Figure 4.8 Bearing load applied to composite for VFM: (a) Superimposed

view of virtual fastener element and bolt-hole interaction; (b)

Components of bearing load applied to model.

88

Figure 4.9 Direction of force exerted on composite material by fastener

and relative position of virtual fastener nodes: (a) no

displacement applied (virtual fastener represented in black), (b)

displacement at Node 1 greater than Node 2 (virtual fastener

represented in green) and (c) displacement at Node 2 greater

than Node 1 (virtual fastener represented in red).

89

Figure 4.10 (a) Displacement approximation technique for VFM applied to a

6 bolt joint; (b) enlarged view illustrating inter-fastener effects

on convergence.

92

Figure 4.11 Graphical Representation of Forced Convergence Procedure. 92

Figure 4.12 Flow chart of Virtual Fastener Method. 93

Figure 4.13 (a) Single column geometry (SC); (b) Double column geometry

(DC) and (c) Triple column geometry (TC). All dimensions are

in mm.

94

Figure 4.14 Student’s t-distribution for sample size of 3 (a) Probability

density function (PDF) and (b) Cumulative distribution function

(CDF).

95

Figure 4.15 Comparison of experimental, direct method and VFM load-

displacement curves for C-layup, single column (SC) joint

configuration (QS=quasi-static, OC=offset correction).

96

xvi

Figure 4.16 Comparison of damage response resulting from randomly

generated fastener properties using the statistical approach.

97

Figure 4.17 Load-displacement curves for multi-fastener joints: (a) C-layup

SC; (a) E-layup SC; (c) C-layup DC; (d) E-layup DC; (e) C-layup

TC. (R) indicates that random fastener properties and (E)

indicates equal properties were used.

99

Figure 4.18 (a) Load-displacement curves for single-bolt C-layup joint

illustrating effect of failure mode on joint strength; (b) Failed SC

C-Layup specimen.

100

Figure 4.19 Load-displacement curves for C-Layup joints with corrected

failure loads; (a) C-Layup SC, (b) C-Layup DC, (c) C-Layup TC.

Numerical curves appended (R) and (E) denote that randomly

generated and equal fastener damage properties were used,

respectively.

101

Figure 4.20 Load-displacement curves of C-layup TC joint with missing

fasteners: (a) MF-9 configuration; (b) MF-8 configuration; (c)

MF-5 configuration; (d) MF-6 configuration. Numerical curves

appended (R) and (E) denote that randomly generated and equal

fastener damage properties were used, respectively.

102

Figure 4.21 Loads carried by individual fasteners in MF-9 Joint: (a) Bolt

numbers and positions; (b) Experimental data (redrawn from

Gray et al (2011)); (c) Direct Method with equal fastener

properties; (d) Direct Method with random fastener properties;

(e) VFM with equal fastener properties.

103

Figure 4.22 CPU Times required for VFM and direct method using constant

displacement increment (2.5µm).

105

Figure 4.23 Effect of displacement step size on the accuracy of the VFM

and direct methods.

105

Figure 4.24 Dialogue boxes to define composite and bolt material

properties.

107

xvii

Figure 4.25 Sections of GUI to define geometry and mesh; (a) laminate

dimensions, seeding and bolt positions, (b) graphic to assist in

definition of dimensions, (c) bolt geometry.

108

Figure 4.26 Finite element analysis solver configuration panel. 108

Chapter 5

Figure 5.1 Comparison of operational times for Direct, Explicit and Virtual

Fastener (VFM) methods: (a) normalised with respect to Direct

method and (b) normalised with respect to Virtual Fastener

method.

113

Figure 5.2 Joint finite element model (a) boundary conditions and

constraints and (b) contact surfaces and user-element locations.

114

Figure 5.3 Acceleration and velocity time histories (a) smooth step; (b)

constant velocity.

115

Figure 5.4 Method for coupling user-element with laminate mesh. 116

Figure 5.5 Simplification of joint region and user-element DOFs. 117

Figure 5.6 a) Bearing load acting on composite material (b) bending

moment in single-shear joint.

118

Figure 5.7 (a) Variation of centre of contact pressure with joint load (b)

countersunk (CSK) and non-countersunk (NCSK) contact

surfaces in a single-bolt, single-lap joint.

119

Figure 5.8 Equivalent spring stiffness model for out-of-plane loading of

joint region.

120

Figure 5.9 Coupling constraint stiffness correction. 121

Figure 5.10 Assumptions regarding recoverable elastic energy and unloading

of joint element.

123

Figure 5.11 Effect of stable time increment on (a) CPU time and (b) Load-

displacement response.

124

xviii

Figure 5.12 Figure 5.12 Modelling of composite laminate using layered,

conventional shell elements.

124

Figure 5.13 Effect of mass-proportional damping on: (a) composite laminate

loaded with a ramp force time-history and (b) load-displacement

response of a single-bolt joint user-element model.

126

Figure 5.14 User-element, single-bolt (SB) joint results: (a) deformed finite

element mesh; (b) load-displacement curve for and (c) out-of-

plane displacement for C-Layup; (d) load-displacement curve for

and (e) out-of-plane displacement for E-Layup. Data series

appended 35% and 50% refer to eccentricity of loading assumed

as a percentage of laminate thickness.

128

Figure 5.15 Internal and kinetic energy in single-bolt E-layup joint. 129

Figure 5.16 User-element results for a single-column (SC) joint: (a) Bolt

numbering convention and virtual strain gauge positions; (b)

Joint load-displacement curve; (c) Comparison of strains output

from virtual strain gauges; (d) Joint load-distribution calculate

from surface strains; (e) Comparison of bolt load distribution

calculated from surface strains on top (TOP) and bottom (BTM)

laminates with force applied through VUEL element

(APPLIED).

131

Figure 5.17 Fastener loads in quasi-static SC C-Lam joint: (a) Load

distribution for joint loaded to ultimate failure load, (b) Detailed

view of load distribution for elastic loading of the joint.

132

Figure 5.18 User-element results for a triple-column (TC) joint: (a) Bolt

numbering convention and virtual strain gauge positions; (b)

Joint load-displacement curve; (c) Load distribution from load

applied through the user-element ; (d) Joint load-distribution

calculated from surface strains (experimental); (e) Joint load-

distribution calculate from surface strains (FE-Model).

133

Figure 5.19 Comparison of surface strains resolved at virtual strain gauge

positions in QS-TC C-Lam joint; (a) between Bolt 1 and 4; (b)

134

xix

between Bolt 2 and 5; (c) between Bolt 3 and 6; (d) between

Bolt 4 and 7; (e) between Bolt 5 and 8; (f) between Bolt 6 and 9.

Figure 5.20 Comparison of raw and filtered SB C-Lam 10m/s load-

displacement curves (a) from VUEL model; (b) redrawn from

Egan at al. (2015).

136

Figure 5.21 Comparison of raw and filtered load-displacement curves

obtained from VUEL model and experimentation by Egan et al.

(2013).

136

Figure 5.22 Load distribution in 9-bolt joint: (a) bolt, row and column

numbering; (b) QS load distribution history; (c) QS load

distribution at 25kN joint load; (d) 5m/s load distribution

history; (e) 5m/s load distribution at 25kN joint load; (f) 10m/s

load distribution history; (g) 10m/s load distribution at 25kN

joint load.

139

Figure 5.23 Propagation of elastic stress wave in C-laminate material arising

from end-displacement loading of 10m/s.

140

Figure 5.24 Time dependent load distribution due to dynamic effects in

triple-column, C-Layup joint loaded at 10m/s: (a) Greatest load

carried by row 1, (b) Load distribution similar to baseline case,

(c) Greatest load carried by row 3.

141

Chapter 6

Figure 6.1 (a) Force “stream tube” and (b) Construction of force

components.

148

Figure 6.2 Effect of fibre orientation on the trajectory of bearing load

paths (a) 0°; (b) 30°; (c) 45°; (d) 60°; (e) 90°.

150

Figure 6.3 Effect of material orientation on load trajectories in multi-

fastener joints: (a) 0°; (b) 90°; (c) 45°; (d) -45°.

153

Figure 6.4 Effect of fibre-orientation on load distribution in a multi-pin

joint.

153

xx

Figure 6.5 Effect of stiffness and anisotropy on load distribution (o) –

orthotropic, (i) – isotropic.

155

Figure 6.6 Comparison of load paths from user-element model and 2D

FEA.

156

Figure 6.7 Load-displacement curves and fastener positions in missing-

fastener cases: (a) MF5, (b) MF6, (c) MF8, (d) MF9.

158

Figure 6.8 Load distribution from VUEL model: (a) MF5; (b) MF6; (c)

MF8 and (d) MF8, Difference in pin load from 2D FEA.

160

Figure 6.9 Patterns used in fastener position study. 162

Figure 6.10 Load distribution in 9-bolt joint configurations. 163

Figure 6.11 Load distribution in 8-bolt joint configurations. 164

Figure 6.12 Distribution of load in 9-bolt joint with fasteners in (a) grid

arrangement and (b) circular arrangement. Relative bolt

positions for (c) grid arrangement and (d) circular arrangement.

166

Chapter 7

Figure 7.1 Application of the developed numerical methods to the

certification procedure for composite materials.

176

Appendices

Figure A.1 Parallelisation benchmark tests for VUEL model. A-8

Figure B.1 Parameters defining load displacement behaviour of a single-

bolt, single-lap joint.

A-10

xxi

LIST OF TABLES

Chapter 3

Table 3.1 Joint Geometries (QS=quasi-static, HS=high speed), All

dimensions are in mm.

69

Table 3.2 Orientations of layups tested. 69

Table 3.3 Averaged key damage variables determined from experimental

data for C and E laminates loaded quasi-statically and at

velocities of 5 and 10 m/s.

69

Table 3.4 Final (catastrophic) failure mode for single bolt cases (FF =

fastener failure, FP = fastener pull through).

70

Table 3.5 Energies associated with non-linear damage (E1) assigned to

each bolt when varying fastener properties were applied to the

quasi-static model

76

Chapter 4

Table 4.1 Number of operations required and complexity of Direct and

Virtual Fastener solution methods.

82

Table 4.2 Mean values (µ) and standard deviations (σ) of damage variables

from single bolt tests (FF=fastener failure, FP=Fastener Pull

Through).

96

Table 4.3 CPU run-times for Direct Method (DM) and Virtual Fastener

Method (VFM) using a single core on a quad-core, 32GB RAM

computer.

106

Chapter 5

Table 5.1 Operations required and complexity of Direct, Virtual Fastener

and Explicit solution methods.

113

Table 5.2 Inertia values for nodes in user-element for composite layups

considered.

118

xxii

Table 5.3 Effect of loading velocity on load distribution in TC C-laminate

joint.

142

Chapter 6

Table 6.1 Initial and ultimate failure loads of multi-fastener configurations. 165

xxiii

NOMENCLATURE

Upper Case

A Constants for cubic curve fit

Ac Projected area of countersunk fastener head onto the shear plane

B (Chapter 3) Constants for conic curve fit

B (Chapter 4) Element strain matrix

C Clearance

D Vector of known control variables

Dde Energy dissipated through damage

DOF Degree of Freedom

E Young’s Modulus

Ec Energy dissipated through viscoelasticity and creep

F Force

FBi Load carried by bolt i

G Shear modulus

I Inertia matrix

Id Number of increments required for direct and VFM analyses

IE Internal energy

Ie Number of increments required for explicit analysis

Ixx, Iyy, Ixy Moments of Inertia

K Stiffness matrix

KB, KBi, Kij Spring stiffness of joint member (i.e. bolt and laminate regions)

Kc Spring stiffness of clamped region in pull-through

Kcpl Equivalent spring stiffness of coupling region

KE Spring stiffness of fastener when loaded via elastic bolt-hole contact

Ke (Chapter 4) Element stiffness matrix

Ke (Chapter 5) Kinetic Energy

KS Spring stiffness of fastener when loaded is reacted through friction

KQLS Spring stiffness value during quasi-linear unloading of joint

L Length of laminate

Mx, My Moment

P Joint Load

PF Ultimate joint load

PFRIC Maximum Joint load reacted through friction

Pi Joint initial failure load

RB Residual force arising from the approximated displacement field

RC Coupling radius

T Transformation matrix

TDM, TVFM Computational times for direct method and VFM

xxiv

TF Force threshold for convergence in VFM

U (Chapter 3) Vector of unknown variables

U (Chapter 4) Displacement field

V, Vx, Vy, Vz Stress pointing vector

Ve Element volume

VFM Virtual Fastener Method

W, Wx, Wy Weight factor for first VFM convergence iteration

XBi Relative displacement of nodes in i-th fastener spring element

XJ End-displacement of joint

XPF Joint displacement at ultimate failure load

XPi Joint displacement at initial failure load

XP0 Joint displacement at zero load

Zp Z-coordinate of centre of contact pressure

Lower Case

a Constant for conic curve fit

c (Chapter 3) Constant for conic curve fit

c (Chapter 5) Damping matrix

cd Dilatational wave speed

d, dh Hole diameter

dH Fastener head diameter

e Edge distance

e (Chapter 5) Distance from shear plane

fNL Non-linear damage function

k Force-per-hole factor

m Mass

n Number of DOF’s in analysis

o Unit vector defining fibre orientation

p, pr, pc Bolt pitch

t Laminate thickness

u Vector of nodal deflection’s

v Vector of nodal velocities

w Laminate width

x Nodal displacement

xB Displacement of fastener node in X-direction

yB Displacement of fastener node in Y-direction

Greek

α Mass proportional damping

β (Chapter 3) Fraction of bending moment reacted by fastener head

β (Chapter 5) Stiffness proportional damping

xxv

Δt Stable time increment

εb Bearing strain

η Convergence increment scaling factor

θ Angle of tangent to conic curve

λ Lamé parameter

μ Shear modulus

ν Poisson’s ratio

ρ Density

σ Stress tensor

σb Bearing stress

τ Shear stress

φ Deviation between local stress trajectory and fibre orientation

Chapter 1

INTRODUCTION

Chapter 1

2

1.1 Background and Motivation

Composites, specifically fibre-reinforced plastics (FRPs), offer a number advantages over

conventional materials such as steel and aluminium alloys. Such advantages include

increased strength-to-weight ratio, corrosion resistance and fatigue life (Niu 1992),

making FRP’s an ideal choice in the manufacture of aerospace and automotive structures.

Consequently composites have seen increased use in light aircraft, military fighters and

helicopters. However, in commercial aviation the introduction of these materials had been

gradual, owing to the high safety standards and the conservative nature of the industry.

One of the first applications of composites in civil aircraft primary structures was in the

1980’s on the 737 horizontal stabiliser (Roeseler et al. 2007). Since then, the use of

composites in aircraft structures have continued to increase with the latest generation of

commercial aircraft, the Boeing 787 and Airbus A350, consisting of over 50% composite

materials by weight (Boeing 2006; EADS 2009). The extensive use of composites in the

Boeing 787 is illustrated in Figure 1.1. Additionally, composite materials have seen

increased use in the automotive industry. For example, carbon fibre reinforced plastic

(CFRP) was used extensively in the manufacture of the Lexus LFA (Toho Tenax 2010),

and in the electric BMW i3 (BMW 2013) and Tesla Roadster (JEC Composites 2011),

both of which feature a 100% CFRP passenger cell. Furthermore, the BMW i3 (formerly

known as the MCV) was the first example of a mass produced carbon fibre based car

(JEC Composites 2011; BMW 2013).

Figure 1.1 Boeing 787 showing breakdown of materials used (redrawn from Georgiadis et al. (2008)).

In contrast to composite materials, the use of mechanical fasteners is a mature technology

and is used to great extent in traditional aircraft structures. This can be seen in the Boeing

747, 767 and 777 which have an estimated 3 million, 1.8 million and 1 million mechanical

Chapter 1

3

fasteners, respectively (Wallace 2008). Although the manufacturing processes associated

with FRPs facilitate an overall reduction in the number of fasteners required,

mechanically fastened joints are still prominent in composite structures. The

predominantly CFRP fuselage of the 787 uses only an estimated 40k-50k fewer fasteners

(Walz 2006) when compared to an aluminium structure of similar size (e.g. 767). This

continued use of bolted joints arises from a number of advantages being associated with

the method, namely less sensitivity to environmental conditions and surface preparation

and the ability to disassemble the joint or detach components for maintenance, repair or

material recycling (Niu 1992).

Bolted joints represent potential weak points in a structure and can limit its overall load

carrying capacity. Therefore it is crucial for design engineers to understand and predict

the behaviour the joint. Joint efficiency is a measure of the strength of the joint compared

to that of the parent material, and in metals values of 70-80% are typical. However, the

efficiency of composite joints is much lower (40-50%), and so optimising the joint design

is essential in order to realise the maximum potential benefits of composites. Some factors

contributing to the lower efficiency in composite joints are the brittle nature of the

material leading to less stress relief around load holes and anisotropy leading to higher

stress concentration factors. In recent years much focus has been placed on the

development of numerical methods to replace time consuming and expensive

experimental tests. The EU FP5 project, BOJCAS – Bolted Joints in Composite Aircraft

Structures was focused on developing advanced numerical methods for bolted joints

(McCarthy 2001). A key technology which resulted from this project was the three-

dimensional finite element (3D-FE) modelling of composite bolted joints. These models

were able to accurately capture the full mechanical behaviour of the joint including effects

of variables such as bolt torque and clearance. Furthermore, using the developed methods,

it was possible to predict damage in the joint on the mesoscale (ply-level). However, a

limitation of this approach was the time required to complete the analyses making 3D-FE

infeasible for large scale studies.

In a report by the National Research Council (2004) to the US department of defence, a

number of recommendations were made regarding the cost-effective, efficient

implementation of new technologies and materials. One such recommendation was a shift

in design philosophy from an analysis-supported testing-based approach to a testing-

supported analysis-based approach with emphasis on efficient modelling and focused

Chapter 1

4

testing. The Defence Advanced Research Projects Agency (DARPA) program on

Accelerated Insertion of Materials - Composites (AIM-C) was used as an example of the

improvements in efficiency that could be made when this philosophy was adopted. Figure

1.2 summarises the achievements of the AIM-C project in terms of successful model

integration as part of the design process. A significant reduction in the time required to

introduce new technology was noted, owed in part to the replacement of a 6-month

experimental series with 2- to 3-day modelling-based activities. Thus, through the use of

robust and high-fidelity numerical models, the time required to implement a new

technology or design can be significantly reduced. Although the AIM-C project was

focused on the introduction of new materials in the design process, the philosophy can be

applied to any new technology including the design of bolted composite structures.

Figure 1.2 Examples of materials and process development acceleration using computational tools under

the AIM-C project (a) traditional testing supported analysis approach (b) analysis approach supported by

experience, testing and demonstration (redrawn from National Research Council (2004)).

The EU FP7 project MAAXIMUS – More Affordable Aircraft through Extended,

Integrated and Mature Numerical sizing was aimed at the efficient development and right-

first-time validation of highly optimised composite fuselage structures (MAAXIMUS

2009). Although one of the aims of the project was the reduction of the amount of

mechanical fastening required, a significant portion of the project was focused on bolted

joint design. This is due to the necessity of bolted joints for inspection and repair purposes

throughout the life-cycle of an aircraft.

Chapter 1

5

Figure 1.3 Certification procedure for composite materials (redrawn from MAAXIMUS (2009)).

The typical experiments required in the certification procedure of composite aircraft

structures are illustrated in Figure 1.3. One of the aims of the BOJCAS project was to

reduce the quantity of experimental tests required during the certification phase through

the use of numerical models (McCarthy 2001). However, given the complexity and

resources required for 3D-FE analyses, this approach was only feasible for coupons and

small sub-assemblies (Levels 1-2 in Figure 1.3). The numerical methods developed

through the MAAXIMUS project allow for simulations of larger structural components

(Levels 3-5) under certain loading conditions. A highly efficient joint model had been

developed for analysis of large composite structures (Gray and McCarthy 2011), however

it was subject to a number of limitations, specifically a high dependence on experimental

data for calibration and an inability to model dynamic problems. The latter is an important

consideration as some critical load cases in aircraft design are dynamic, i.e. bird-strike

and tyre-debris impact.

Chapter 1

6

1.2 Objectives

The primary objective of this thesis is the development of a highly efficient numerical

method for modelling composite bolted joints at static and dynamic loading rates. For

impact and crash analysis the behaviour of the joint beyond initial failure and up to

complete separation is required (Gunnion et al. 2006), and thus any method developed

must be able to account for this. Furthermore, to achieve maximum effectiveness for use

in industry the method developed should also be applicable to the testing-supported

analysis-based design philosophy. This requires highly accurate numerical models which

allow the joint design variables (such as clearance, bolt torque, strength, etc.) to be easily

varied. A certain degree of experimentation will always be necessary as part of the design

and certification process of aircraft. However, it is envisioned that results from coupon

testing (Level 1 in Figure 1.3) be used only to calibrate highly efficient numerical models

which are then used in lieu of experiments in Levels 2-5. Therefore, regarding the

development of an efficient analysis tool, the primary objectives are as follows:

1) Investigate and compare numerical solution procedures in terms of computational

efficiency. The developed model should use the most efficient solution

methodology available.

2) Develop a representative model for bolted joints capable of capturing any

dynamic loading effects and modelling the joint to complete separation.

3) Integrate a representative model for bolted joint in commercial FE software to

provide a robust, user-friendly tool for industry.

Additionally, there are a number of factors regarding the behaviour of bolted joints that,

due to limitations of experimental methods and traditional modelling techniques, have

not been fully investigated. Once a highly efficient and accurate joint model has been

developed, the following secondary objectives are set:

1) Investigate the effect of statistical variations of fastener properties in multi-bolt

joints.

2) Investigate factors affecting load distribution in multi-fastener joints. Parameters

to consider are material properties and layup, loading velocity and the relative

spacing between fasteners.

3) Based on the findings of numerical studies, propose a series of guidelines for the

design of multi-fastener composite joints.

Chapter 1

7

1.3 Thesis Outline

In Chapter 2 a review of relevant publications was conducted. Firstly, the literature

pertaining to joint design variables and how these effect the mechanical response of the

joint was discussed. Such variables include bolt torque, clearance, composite layup and

loading rate. Secondly literature relating to detailed finite element modelling of bolted

joints was reviewed. The purpose of this was to determine the capabilities and limitations

of using traditional modelling techniques. Finally, a review was conducted on efficient

numerical modelling strategies to determine the current state-of-the-art. Areas of focus in

this section include purpose finite elements and analytical models.

Chapter 3 presents a semi-empirical method for approximating the damaged response of

shear-loaded composite bolted joints. Using a novel application of a conic function, the

damaged load-displacement behaviour of the joint was approximated knowing only the

initial and ultimate failure loads and energy absorbed. When applied in a simple

equivalent spring model accurate load-displacement curves to failure of single-row,

multi-fastener joint configurations were obtained. This method was validated against

experimental data and was found to provide accurate results for both static and dynamic

loading conditions.

In Chapter 4, the analytical model presented in Chapter 3 was applied to a custom, two-

dimensional finite element code with a novel highly efficient solver methodology. This

allowed for the highly efficient analysis of two-dimensional fastener arrays. Using the

parameterised damage approximation function it was possible to vary the damaged

response at each fastener location. Using this approach, the statistical variations in energy

absorption and failure loads that were observed experimentally were applied to the finite

element model to investigate what effect this had in multi-fastener joints.

Chapter 5 presents the development of a highly efficient user-defined finite element for

modelling the three-dimensional behaviour of single-shear joints. The element was easily

implemented in commercial FE software and was validated against experimental data and

results from detailed three-dimensional FE analyses. The element is capable of capturing

the full non-linear shear load-displacement behaviour of the joint, in addition to

accounting for the stiffness of the clamped region when loaded in the transverse direction.

The element was developed specifically for use in an explicit FE solver, making it

particularly suited for dynamic problems.

Chapter 1

8

In Chapter 6, the effect of fibre orientation and material anisotropy on load distribution

in multi-fastener joints was investigated. Using a novel qualitative visualisation method

trajectories of bearing load were illustrated. The user-element developed in Chapter 5 was

also applied in this chapter to investigate the effects of missing fasteners and relative bolt

spacing on load distribution in multi-fastener joints.

Finally, in Chapter 7 concluding remarks and recommendations for future work are

presented. The methods developed in this thesis were discussed in the context of their

potential applications in relation to the certification of composite aircraft. The findings

made in Chapters 3, 4, 5 and 6 regarding the mechanical response of bolted joints are

summarised and a number of design recommendations are presented for multi-fastener

joints.

Chapter 2

LITERATURE REVIEW

Chapter 2

10

2.1 Introduction

Bolted joints are an important consideration in the design of both composite and

metallic structures. Consequently, extensive research has been carried out on this topic

over the last number of decades. Mechanical behaviour, development of experimental

techniques and the development of numerical and analytical design tools are aspects

which have been investigated.

This literature review is divided into three main parts. Firstly, publications pertaining

to the mechanical response of bolted joints are discussed. Following an introduction

to the terminology related to bolted joints, a description of common failure modes is

provided. Publications which discuss the effects of design variables such as geometry,

composite layup, clearance and loading rate are then reviewed. The second section

provides a brief discussion on the capabilities of three-dimensional finite element (3D-

FE) analyses of bolted joints. Specifically, the ability to capture both the undamaged

response of the joint in addition to the prediction and simulation of joint failure was

examined. Finally, efficient modelling strategies for bolted joints are reviewed.

Analytical models in addition to highly simplified global modelling techniques were

discussed. To supplement this review of literature, the main points from each section

are summarised at the end of the chapter.

2.2 Mechanical Behaviour of Bolted Joints

Due to the interaction of multiple components and a significant number of variables

associated with the problem, predicting the behaviour of bolted joints is a non-trivial

task. A large number of experimental and numerical studies have been carried out to

investigate the influence of various factors on the response of composite bolted joints.

This section provides a summary of these studies in addition to an introduction to the

terminology relevant to bolted joints.

2.2.1 Terminology

In this section, terms pertaining to the geometric configuration, loading, and failure of

bolted joints are presented. Figure 2.1 illustrates a number of commonly used

geometric parameters. The length of the laminate, L, consists of two regions, the

overlap and the non-overlap region. The overlap region is the section of laminate in

contact with another laminate, and this interface is referred to as the shear plane. It is

Chapter 2

11

possible for the joint to have more than one shear plane, as is the case for the double-

lap joint in Figure 2.1. However, the single shear bearing configuration is

representative of most aircraft bolted joint applications (MIL-HDBK-17 2002). This

is due to the bending and shear loads which are induced on the fastener, while the

double-lap joint induces mostly shear loads. Hole diameter, width and edge distance

are defined as d, w, and e respectively while laminate thickness is denoted by the

variable, t. In multi-fastener joints, two additional parameters are used to define the

position of the bolts relative to each other. These are the distances between the hole-

centres along the loading direction, pr (row pitch), and transverse to the loading

direction, pc (column pitch). For the entirety of this thesis, “rows” refer to fasteners

groups aligned with the main loading direction while “columns” are oriented

transverse. To assist in the explanation of failure modes in bolted joints, failure planes

have been illustrated in Figure 2.1, these include the bearing plane, net-tension plane

and shear-out plane. Failure modes in bolted joints will be discussed in detail in

Section 2.2.2.

Figure 2.1 Definition of geometric parameters of bolted joints (redrawn from McCarthy (2003)).

A number of additional terms are used to describe the strength and efficiency of joints.

Bolted joints are often analysed in terms of load-displacement curves or bearing stress-

strain curves. The displacement undergone by the joint is typically taken as the

crosshead displacement of the test machine, unless extensometers are fitted to the

specimen. The total load carried by the joint is taken from a load-cell affixed to one of

the crossheads. There is no standard method to determine the loads carried by

individual bolts, however some methods include the use of instrumented fasteners

Chapter 2

12

(Lawlor 2004; Ekh and Schön 2006) and from the measurement of surface strains

(Starikov and Schön 2002; Lawlor et al. 2005). These will be discussed in more depth

in Section 2.2.9.

To determine bearing strength, bearing stress-strain curves may be derived. From

MIL-HDBK-17 (MIL-HDBK-17 2002) relating to the characterisation of polymer-

matrix composites and from ASTM Standard D5961/D5961M-08 (ASTM 2008) on

testing the bearing response of composite materials, in a single bolt joint, bearing stress

is defined as:

dt

Pb (Eq 2.1)

Where P is the load carried by the joint. Bearing strain is defined as the ratio of the

deformation of the bearing hole in the direction of the applied force to the pin diameter

is defined as:

dk

b21

(Eq 2.2)

Where δ1 and δ2 are the displacements measured at extensometers 1 and 2 respectively,

and k is the “force-per-hole factor”, set to either 1.0 for single-lap or 2.0 for double-

lap joints. A final term used in the discussion of bolted joints is the joint efficiency

which is defined by Eq. (2.3) (Hart-Smith 2003). This is the ratio of joint strength to

that of the parent base material.

StrengthLaminate notched-Un

StrengthJoint Efficency Joint (Eq 2.3)

2.2.2 Failure Modes of Bolted Joints

When designing a joint, or any structure, it is important to consider the modes of

failure which it can undergo. The six main failure modes in bolted joints, are illustrated

in Figure 2.2. It is important to note that the occurrence these failure modes are not

random and have been found to depend strongly on geometry and layup. The

dependence of failure modes on these parameters is the subject of Sections 2.2.3 to

2.2.7. However, in this section a brief description of each failure mode is provided in

addition to a short discussion on their desirability and the micromechanical

mechanisms leading to failure.

Chapter 2

13

Bearing failure is a progressive failure mode that occurs in the material immediately

adjacent to the points of contact between the fastener and the laminate. It is thought to

occur when the ratio of d/w is low or the ratio of by-pass to bearing load is small.

Bearing failure was also found to be strongly affected by clamping pressure (Eriksson

et al. 1995). A characteristic of this failure mode is the non-linear load displacement

behaviour before final failure which arises from the accumulation of damage.

Camanho et al. (1998) found bearing failure to initiate with localised delamination.

As load increases matrix cracking occurs in the 90° and 45° plies with these cracks

providing preferential starting points for other damage mechanisms. Fibre micro-

buckling was also observed to occur in 0° plies and appeared to be related to the

presence of matrix cracks in the 45° plies.

Figure 2.2 Failure modes of mechanically fastened joints (redrawn from Niu (1992)).

Net-tension failure is a catastrophic failure mode which occurs without significant

warning. This mode of failure is most likely to occur when the ratio of d/w is high, or

when the ratio of by-pass to bearing load is high (Eriksson et al. 1995). Camanho et

al. (1998) found no significant damage in joints designed to fail by this mode until

Chapter 2

14

90% of the ultimate failure load was carried. It was at this load that delamination in

the 90°/45° interfaces began to occur. Additionally matrix cracking in the off-axis

plies was present and lead to non-linearity in the load-displacement curve.

Shear-out failure is another catastrophic mode that provides little warning before the

complete failure of the joint. Typically characterised by a linear load-displacement

relationship until final failure (Camanho et al. 1998), shear-out is prone to occur in

highly orthotropic laminates and those with short end distances (Eriksson et al. 1995).

Similar to net-tension failure, Camanho et al. (1998) found damage to occur as late as

90% of the ultimate failure load, when at this stage crushing and delamination became

noticeable. Cleavage failure is a combination of the net-tension and shear-out modes

and occurs due to the proximity of the hole to the end of the specimen (Hart-Smith

2003; MIL-HDBK-17 2002). This type of failure often initiates at the end of the

specimen rather than adjacent to the fastener (MIL-HDBK-17 2002).

Fastener pull-through is an out-of-plane failure mode which is characterised by a linear

load-displacement response up to the ultimate load (Waters and Williams 1985;

Banbury and Kelly 1999; Kelly and Hallström 2005). Through-thickness failure at the

edge of the fastener head and subsequent in-plane delamination are the damage

mechanisms characterised by this mode of failure (Banbury and Kelly 1999). Fastener

pull-through is highly dependent on the fastener, being frequently associated with

countersunk joints (MIL-HDBK-17 2002). Resistance to pull-through failure can be

increased by increasing the fastener head diameter (Banbury and Kelly 1999).

Fastener failure is seen as a premature mode of failure and is generally considered

undesirable. McCarthy et al. (2002) observed joints which initially failed in bearing

ultimately failed due to fastener failure. Egan et al. (2013) also observed bolt failure

in single-lap, countersunk joints which was attributed to a large bending moment

reacted by the fastener head as a result of the thickness of the laminate.

Of the failure modes illustrated in Figure 2.2 only bearing and net-tension are

considered desirable, as all other modes result in premature failure (Hart-Smith 2003).

It was found by Hart-Smith (2003) that joints designed to fail via net tension tend to

carry the greatest loads. Despite this, joints are typically designed to initially fail in

bearing as this is a progressive mode, allowing damage to be detected before

catastrophic failure of the joint occurs.

Chapter 2

15

2.2.3 Material and Lay-up

A well-known characteristic of composite bolted joints is a lower structural efficiency

when compared to their metallic counterparts. For comparison, optimal designs of

composite joints rarely exceed a structural efficiency of 40%. However joints

manufactured from ductile metals such as aluminium can reach in excess of 60%. The

reduction in efficiency is generally attributed to higher stress concentrations associated

with orthotropic materials (Collings 1977; Hart-Smith 2003). When only structural

efficiency is considered, there is a clear advantage to using conventional materials.

However, Collings (1977) compared the ratios of specific strengths of CFRP and

metallic joints and it was observed that composites offer a potential advantage of 32-

52% when compared to aluminium alloys and 79-107% for steel. Thus when selecting

composites over conventional materials, structural efficiency in the joint region is

sacrificed for a superior strength-to-weight ratio.

Fibre orientation in multidirectional laminates has been found to play an important

role in determining the failure mode in bolted joints. Collings (1977) found that large

numbers of ±45° plies could reduce the stress concentration around the hole by

imparting a degree of softening in the joint. Arnold et al. (1990) also emphasised the

importance of 45° plies as a mechanism of diffusing loads around the bolt hole, while

Eriksson (1990) found laminates that used higher proportions of ±45° plies yielded

higher bearing strength. Experiments from Collings (1977) suggest that the best tensile

performance is achieved when using 30-50% 45° plies in [0°/ ±45°] laminates.

Furthermore, Kelly and Hallström (2004) observed that a high percentage of ±45°

acted to inhibit pure shear-out failure in the joint sections tested. From a review by

Hart-Smith (2003), it was noted that shear-out and cleavage failures can arise, in part,

due to a highly orthotropic laminate pattern or if there is insufficient dispersion of the

differently oriented plies. It was recommended that fibre patterns be fully dispersed to

maximise resin interfaces between changes in direction of layers of fibres.

Orientations in a multidirectional laminate should consist of at least 12.5% of plies in

each of the four directions: 0°, ±45° and 90° (Arnold et al. 1990; Hart-Smith 2003).

Both Wang et al. (1996) and Park (2001) found joints with 90° plies on the surface of

the laminate had greater delamination bearing strengths than those with 0°. This

resulted from the tendency of 0° plies located on the surface to fail by breaking away

from the laminate due to bearing loads (Wang et al. 1996).

Chapter 2

16

A benefit of the use of multidirectional laminates is the ability to tailor the orthotropic

material properties to suit the loading conditions that the structure is expected to

experience. This primarily consists of orientating a large proportion of 0° in the

primary loading direction. Although experiments from Pakdil et al. (2007) found that

bearing strength was maximum in zero-dominated layups, evidence from the majority

of sources reviewed suggest that a quasi-isotopic layup is preferential to optimise the

bearing response of laminates (Collings 1977; Hart-Smith 2003). Furthermore, Hart-

Smith (2003) noted that joint strength varies less with the percentage of 0° plies than

does the notched laminate strength.

2.2.4 Geometry

It is apparent that the performance of a joint is strongly governed by its geometry. This

section reviews a number of published works which discussed the influence of various

geometric ratios on ultimate bearing strengths and failure modes of bolted joints. Other

important geometric parameters such as bolt-hole clearance and multiple fasteners are

discussed in Sections 2.2.6 and 2.2.9 respectively.

The effect of the w/d and e/d ratios was investigated by Cooper and Turvey (1995),

Collings (1977), Hart-Smith (2003) , Kelly and Hallström (2004) and Pakdil et al.

(2007). All sources found that increasing the w/d ratio resulted in a change in failure

mode from net-tension to the more progressive bearing failure. However, the w/d ratio

at which this transition occurs was observed to vary with layup and clamping pressure.

Collings (1977) found the transition to bearing failure occurred at w/d ratios between

3.5-4 in layups consisting of [0°/±45°], [0°/±60°] and [0°/90°] plies but at 7 for ±45°

laminates. Kelly and Hallström (2004) found the critical w/d ratio for the transition to

bearing failure was lower in pinned joints (w/d = 2) compared to clamped joints (w/d

= 3). This was attributed to the lateral support at the edge of the hole inhibiting through

thickness expansion.

Increasing the e/d ratio was also found to result in a change in failure mode, where

shear-out failure occurred at lower ratios while larger ratios promoted bearing failure

(Collings 1977; Cooper and Turvey 1995; Kelly and Hallström 2004). Again, the

critical e/d ratio resulting in this change in failure mode was found to be dependent on

layup and clamping pressure, with the larger e/d ratios required in clamped joints

(Kelly and Hallström 2004). The orientations which resulted in the lowest ratios were

Chapter 2

17

[0°/±45°], [0°/±60°] requiring an e/d ratio of 2.5-3 to promote bearing failure (Collings

1977). For the joints tested by Kelly and Hallström (2004), it was found that once

sufficiently large e/d and w/d ratios were used so that bearing failure occurred, further

increase in these ratios resulted in very little change in the ultimate strength of the

joint. This suggests that the ultimate strength of the joint is governed by the failure

mode. Similar findings were made by Cooper and Turvey (1995).

The effect of laminate thickness was also discussed in literature. Eriksson (1990),

Kelly and Hallström (2004), Collings (1977) and Egan et al. (2013) found an increase

in ultimate bearing strength with increasing t/d ratio. Additionally Egan et al. (2013)

noticed a change in failure mode from one of fastener pull through to fastener fracture

with increasing thickness. This was attributed to a greater bending moment reacted by

the fastener head in thicker layups. However, the use of too small a bolt diameter

results in excessive bolt bending and a highly non-uniform bending stress distribution

which tends to promote pull-through failure (Hart-Smith 2003). To further the

argument of using larger diameter fasteners, both Eriksson (1990) and Wang et al.

(1996) found an increase in ultimate bearing stress with increasing bolt diameter, using

a constant w/d ratio.

2.2.5 Fastener Type

Few authors have directly compared the response of joints fastened with protruding

head and countersunk bolts. McCarthy et al. (2002) noted a reduction in stiffness in

countersunk joints when compared to their protruding head counterparts. Furthermore,

a 25-35% reduction in bearing strength was observed which was thought to be due to

the presence of high stress concentrations between the bolt shank and hole. Hart-Smith

(2003) also noticed these trends, citing the reduced effective bearing area in

countersunk joints as a contributing factor.

The removal of material to accommodate a countersunk head also has a significant

effect on the failure behaviour of the joint. Gunnion et al. (2006) found the primary

difference between the responses of the two head types was that protruding head

fasteners exhibited a sudden transition from the linear elastic to damage region. In

comparison this transition was much more gradual in countersunk specimens. Pearce

(2009) observed different failure modes for the two head types. Countersunk joints

were observed to fail via bolt failure at quasi-static rates but via bearing failure at

Chapter 2

18

higher rates. In comparison, protruding head joints failed via net-tension and cleavage

when loaded quasi-statically and bearing-cleavage at higher rates. Of course these

observations are specific to the material and geometry used by Pearce (2009), but they

highlight the effect of fastener type on the failure response of the joint. In general,

countersunk joints are more prone to bearing failure and fastener pull-through (Niu

1992).

2.2.6 Bolt-hole Clearance

In any manufacturing process, there is a trade-off between maximising quality and

minimising process costs. As a consequence of this, statistical variations from nominal

hole and bolt diameters are to be expected. The difference between these values results

in bolt-hole clearance, which can significantly affect the contact angle (see Figure 2.3).

Clearance can also be expressed as a percentage as shown in Figure 2.3 (a), and values

of 1.2% are typical of aircraft joints (Crews and Naik 1987). Numerous experimental

and numerical studies have been carried out to determine the influence of clearance

on the performance of composite bolted joints.

Naik and Crews (1986) carried out 2D finite element (FE) analysis to determine the

effect of clearance in pin-loaded composite joints. The FE model developed was

compared to the continuum analysis of Eshwar (1977), which allowed the contact

angle to be predicted as a function of pin load, and excellent agreement was observed

for the case of an infinite isotropic plate. Clearances ranging from 0-1.6% were

investigated by Naik and Crews (1986). For 1.6% clearance, peak radial and tangential

stresses were found to increase by 36% and 16% respectively over the neat-fit case.

The contact angle was found to decrease by 30% and 16% greater hole elongation was

observed. It was concluded that clearance in mechanically fastened joints should be

considered in stress and strength analyses but may have little influence on joint

stiffness.

DiNicola and Fantle (1993) tested double shear pin joints with clearances ranging from

neat-fit to 279µm in addition to carrying out FE analyses. It was found that bearing

strength at four percent hole deformation depended strongly on bolt-hole clearance,

with the 279µm clearance case exhibiting a 30% reduction in strength. As expected,

the distribution of, and peak contact forces between the pin and hole was greatly

Chapter 2

19

influenced by hole oversize. However, ultimate bearing stress of the joint was not

found to be dependent on clearance.

Figure 2.3 Definition of (a) clearance and (b) contact angle.

Lawlor et al. (2002a) carried out experimental studies on single-shear, single-bolt

graphite/epoxy joints with quasi-isotropic and zero-dominated layups. Nominal

clearances of 0, 1, 2 and 3% were investigated. It was found that for finger-tight,

protruding head bolts in quasi-isotropic layups there was a delay in load take up, which

is slightly larger than the nominal clearance value. Similar findings were made for

countersunk fasteners, although the repeatability in load take up for various clearances

were not as good as those observed for protruding head bolts. In highly torqued joints,

the clearance was found to be taken up after the effects of static friction had been

overcome. All specimens considered initially failed in bearing, however greater

ultimate strain was observed with increasing clearance. It was believed this increase

in ultimate strain occurred as larger clearances resulted in more concentrated loads on

the laminate, thus resulting in extensive laminate damage. As the laminate absorbs

more energy, there is this less remaining to be absorbed by plastic deformation of the

bolt. Kelly and Hallström (2004) also observed extensive bearing damage in joints

with larger bolt-hole clearance. Similar to DiNicola and Fantle (1993), Lawlor et al.

(2002a) and McCarthy et al. (2002), Kelly and Hallström (2004) found the effect of

clearance on the ultimate strength of the joint to be negligible. In contrast, Pierron et

al. (2000), found ultimate failure loads to decrease by approximately 30% in pin-

Chapter 2

20

loaded, woven glass fibre epoxy joints. However, this reduction in strength

corresponded to a clearance of 12.5%.

In Lawlor et al. (2002a), a reduction in joint stiffness was observed when clearance

was increased. Comparing the joint stiffness of the neat-fit and maximum clearance

cases, a 10% decrease in stiffness was noted. McCarthy et al. (2002) found that the

reduction in stiffness in 1% clearance joints with protruding head fasteners was

statistically insignificant, however in countersunk joints a stiffness reduction of

approximately 5% was noted for the same clearance value. For 2% clearance, a 7-10%

reduction in stiffness was observed for protruding head fasteners and 3-4% for

countersunk. A reduction in joint stiffness in single-bolt clearance fit joints was also

observed by McCarthy and McCarthy (2003) in 3D-FE models and experimentally by

Kelly and Hallström (2004). In contrast, Naik and Crews (1986) and Pierron et al.

(2000) concluded that clearance had little effect on joint stiffness, however this was

not quantified by either author.

Clearance in double-lap, multi-bolt joints was investigated experimentally by Lawlor

et al. (2005). Using strain gauges positioned on the laminate surface, the effects of

clearance on bypass loads were measured. It was found that bolts with larger

clearances do not initially carry any load, resulting in the remaining fasteners carrying

the majority of the joint load. Once the distance between the bolt and hole has been

taken up, larger clearance fasteners begin to contribute to the load carrying capacity

of the joint and the load distribution started to even out. McCarthy and McCarthy

(2003) found similar results in their numerical models. However, Lawlor et al. (2005)

found this process was to be interrupted by the onset of material failures. Clearance

was discovered to have a significant effect on the initial failure loads in multi-bolt

joints, i.e. the load at which bearing failure occurs in one of the holes. The initial

failure load in a 3-bolt joint was found to decrease by 25% with non-uniform

clearances at each hole, when compared to the neat fit case. However, as in single-bolt

joints, clearance was not found to affect the ultimate load carried in multi-bolt

configurations.

2.2.7 Bolt Torque and Lateral Constraint

The clamping effect that arises due to bolt torque can have a significant influence on

the joint response. A number of authors have investigated the effect of lateral

Chapter 2

21

constraint and clamping pressure in bolted joints. In the work carried out by McCarthy

et al. (2002), the effect of bolt torque was investigated. Two different torque levels

were considered, 0.5 Nm to represent “finger tight” conditions and 16 Nm as the

recommended in-service torque. In “finger tight” joints a distinct “knee” was observed

in the load-deflection curves of quasi-isotropic layups. Based on the findings of Park

(2001), and Kelly and Hallström (2004), this was most likely due to the onset of

bearing damage, which explains why this occurrence was most noticeable in

countersunk joints. In zero-dominated layups and highly torqued joints this knee was

not evident. In highly torqued joints, a distinct transition from an initial high slope to

a lower slope was noticed, with the length of the transition region approximately equal

to the clearance in the joint, as illustrated in Figure 2.4. The initially high stiffness of

Slope 1 was believed to be dominated by static friction, and the lower slope during the

transition region from kinetic friction. McCarthy et al. (2002) noted that the linearity

of this high stiffness region was debatable, however assuming only static friction to

be dominant McCarthy and Gray (2011) developed an analytical model for highly

torqued, multi-bolt joints with clearance. Excellent correlation was observed between

the analytical model, experiments and 3D-FE analyses. In Figure 2.4, Slope 2 was due

to the stiffness of the untorqued joint in addition to kinetic friction.

Figure 2.4 Effect of bolt torque on initial load-displacement curve of single-bolt, single-lap joint with

clearance (redrawn from McCarthy et al. (2002)).

Using 3D-FE analyses, McCarthy et al. (2005a) modelled the behaviour of single-lap,

protruding head, composite bolted joints manufactured from HTA/6376. Bolt torque

was simulated by applying orthotropic thermal expansion to one of the washers, and

friction from a stick-slip model. This stick slip friction model attempted to capture the

Chapter 2

22

discontinuity in the idealised Coulomb model through a series of conditional

statements.

Schön (2004a) carried out a series of experiments on HTA/6376 and found the

coefficient of friction for composite to composite contact varied between 0.65 and

0.74. The same author (Schön 2004b), also found the coefficient of friction between

composite and aluminium to have an initial value of 0.23 and peak value of 0.68 after

wear in. Tsukizoe and Ohmae (1986) found that the coefficient of friction between

mild steel and unidirectional carbon fibre-reinforced epoxy was dependent on the fibre

volume fraction and varied between 0.1 and 0.3. Based on these publications,

McCarthy et al. (2005a), used the following coefficients of friction: µ=0.7 for laminate

to laminate, µ=0.3, for washer (steel)-to laminate and µ=0.1 for bolt (titanium) to

laminate contact. Egan et al. (2015) used these same properties in the modelling of

dynamically loaded countersunk M21E/IMA composite joints. The stick slip

algorithm used in McCarthy et al. (2005a) required additional numerical parameters

which were tuned to allow convergence of the finite element model and to provide

accurate results.

McCarthy et al. (2005a) found that Slope 1 in Figure 2.4 arose as the load carried by

the joint was almost entirely reacted by friction between the laminates. During the

transition region, friction between the washer and laminate which previously had little

effect on the response of the joint now account for approximately one third of the load

carried. For Slope 2, the load carried by friction between the laminates drops slightly

with the greatest portion of the load reacted by the normal contact between the bolt

and hole. Friction between the fastener and hole was found to have a negligible effect

on the load displacement history. This is best illustrated in Figure 2.5 (a). It was also

found that after the effects of friction have been overcome and the majority of the load

was carried by the contact between the bolt and hole, the total friction load starts to

reduce. The reason given for this is that for single lap joints, as a consequence of bolt

rotation, the plates start to pull apart resulting in a loss in normal load and hence

friction force between the plates. This effect was also found to be dependent on hole

clearance and is illustrated in Figure 2.5 (b).

Chapter 2

23

Figure 2.5 (a) Loads carried by various interactions in single-lap joint (b) Bolt and total friction load

distribution in multi-bolt joint with variable hole clearance, C1=neat-fit, C4=240 µm clearance

(redrawn from McCarthy et al. (2005a)).

In a study carried out by Park (2001), clamping pressure was found to supress the

onset of delamination around the hole in bolted joints, and as a result delamination

failure tended to occur outside the clamped region. As a consequence of the

suppression of delamination, the bearing failure which was observed to be a

catastrophic failure mode in pinned joints, was progressive in bolted joints. It was

noted that the ultimate and delamination strengths of the bolted joint were nearly

double that of the pinned joints. Similar results were obtained by Kelly and Hallström

(2004) where a number of pinned and bolted composite joints were tested

experimentally. It was found that the lateral constraint imposed on the composite

laminate by the fastener tended to supress a “brooming” type failure at the edge of the

hole observed in pinned joints. The effect of lateral constraint on joint strength is

illustrated in Figure 2.6 (a). Eriksson (1990), Cooper and Turvey (1995), Khashaba et

al. (2006) and Pakdil et al. (2007) also found the bearing strength of the joint to

increase with bolt torque.

In a study carried out by Park (2001), it was found that above a nominal clamping

pressure, there was little increase in the ultimate bearing strength of bolted joints. This

is illustrated in Figure 2.6 (b) for three different composite layups. Furthermore it was

noted that at a clamping pressure of approximately 0.1% of the nominal bolt torque,

the ultimate bearing strength was almost double compared to that for pinned joints.

These results suggest that the increase in strength of bolted joints over pinned joints

was primarily due to the lateral constraint of the fastener, rather than the increased

friction between the laminates. This was also observed by Nassar et al. (2007) in

double-lap, composite-aluminium joints. Further evidence of this can be seen in Figure

Chapter 2

24

2.5 (b) where McCarthy et al. (2005a) showed that the friction in the joint decreases

as loading progresses. However, Park (2001) also noted that although increasing

clamping pressure above a nominal value did not significantly change the ultimate

bearing strength of the joint, the delamination bearing strength continued to increase

with increasing clamping pressure. In contrast to Park (2001), McCarthy and Gray

(2011) found that joints with larger values of torque carried greater ultimate loads.

This was attributed to the onset of critical bearing damage occurring at higher load

levels due to friction forces at the shear plane.

Figure 2.6 (a) Load-displacement behaviour of pinned and clamped laminated (redrawn from Kelly

and Hallström (2004)); (b) Effect of clamping pressure on ultimate bearing strength of bolted joints

(redrawn from Park (2001)).

2.2.8 Loading Rate

A vast number of experimental studies have been carried out to investigate the strain

rate effects in materials and mechanically fastened structures. This section provides a

brief summary of the studies of relevance to the subject matter of this thesis. Firstly a

summary on the findings of rate effects in the parent material, specifically the

composite is given followed by experimental findings in bolted joints.

The effect of strain rate on the mechanical behaviour of composite materials has been

investigated by a number of authors using various experimental techniques. Using

drop-tower testing and a servohydraulic test machine, Hsiao and Daniel (1998)

investigated the rate effects on the compressive and shear behaviour of carbon/epoxy

composites for strain rates up to 300s-1. A significant enhancement in matrix-

dominated properties was observed for higher loading rates. Shear modulus and

strength were found to increase by 18% and 80% respectively, while the transverse

Chapter 2

25

modulus and strength increased by 37% and 100%. The ultimate strain at failure for

both shear and transverse loads appeared to be rate independent. Figure 2.7 illustrates

the effects on strain rate observed by Hsiao and Daniel (1998) in matrix-dominated

properties. In a similar study carried out by Hsiao et al. (1998), the variation of both

strength and modulus with strain rate was investigated further. Between quasi-static

and rates of 10s-1, a linear dependence on strength/modulus was observed with the log

of strain rate. However, for strain rates above 10s-1, strength/modulus appeared to

increase exponentially. When the longitudinal properties were investigated only a

slight increase in the compressive modulus was reported, however strength and

ultimate strain values increased by up to 79% and 74% respectively. For strain rates

up to approximately 100s-1, only a linear variation in strength with the log of strain

rate was observed.

Using a Split Hopkinson Pressure Bar (SHPB), Koerber et al. (2010) investigated the

strain rate effects on the transverse compression and in-plane shear properties of

carbon-epoxy composites for strain rates up to 350s-1. The trends observed in terms of

modulus and strength were comparable to those observed by Hsiao and Daniel (1998).

Using SHPB testing Hosur et al. (2001) investigated the high strain rate compression

response on unidirectional and cross-ply carbon epoxy laminates up to rates of 817s-1.

For all cases tested, an increase in stiffness was observed when loading rate was

increased from quasi-static to dynamic. However, in cross-ply laminates (loaded in

plane and through thickness) and unidirectional laminates loaded along 0°, a decrease

in stiffness was observed when strain rate was increased from 163s-1 to 817s-1. A

decrease in peak stress was also noted in the case of cross-ply laminates loaded in-

plane. Additionally, the peak stress in all cross-ply laminates loaded dynamically

through the thickness was lower than the equivalent quasi-static case. However, the

behaviour of the 90° plies was similar to that observed in Hsiao and Daniel (1998) and

Koerber et al. (2010), in that both stiffness and strength were seen to increase with

strain rate.

Chapter 2

26

Figure 2.7 Quasi-static and high rate stress strain curves for: (a) Transverse compression and; (b)

shear in unidirectional carbon/epoxy composites (redrawn from Hsiao and Daniel (1998)).

Groves et al. (1992) tested carbon/epoxy composites in tension, shear and

compression up to rates of 3000s-1. Their findings supported those of Hsiao et al.

(1998), specifically the exponential-like increase in stiffness and strength beyond

strain rates of 10s-1. Several theories were postulated to explain the rate-dependent

behaviour observed in polymer-matrix composites. Hsiao and Daniel (1998) proposed

that the time-dependent nature of accumulating damage could be one source of the

strain rate effects observed. At lower rates, damage accumulates more gradually, such

that a well-defined non-linear region occurs near the end of the stress