Design of Composite Structures Containing
Bolt Holes and Open Holes
by
Tomas Ireman
Department of Aeronautics
Kungliga Tekniska Högskolan
(Royal Institute of Technology)
SE-100 44 Stockholm
Sweden
Report No. 99-03
ISSN 0280-4646
PREFACE
The work presented in this doctoral thesis was carried out between January 1991 and October
1998 at Saab AB and at the Royal Institute of Technology, Department of Aeronautics. The
work was carried out on part time basis. The thesis consists of five separate papers, of which
two were a part of my licentiate thesis presented in December 1994.
The work was financially supported by Saab AB and the Swedish National Aeronautical
Forum.
First I would like to thank my supervisor, Dr. Ingvar Eriksson, for his support and
encouragement throughout this study and Professor Jan Bäcklund for giving me the
opportunity to carry out this work within the department.
I also want to thank all my colleagues at Saab AB, and especially my former heads Lars
Sjöström and Hans Ansell for their support and Christina Altkvist, Anders Bredberg, and
Tonny Nyman for their helpfulness and for many hours of stimulating discussions.
Finally, I thank my wife Christina Bradley Ireman and my children Emilie, Amanda, and
Wilhelm for their patience and support during the course of this study.
Linköping, January 1999
Tomas Ireman
i
ABSTRACT
This thesis is concerned with stress analysis and strength prediction in composite laminates
containing bolt holes and open holes. It covers both two- and three-dimensional analyses, and
experiments have been carried out to determine criteria parameters and to validate the analysis
methods.
The objective of the work on two-dimensional analysis methods has been to develop accurate
and cost-efficient stress analysis and failure prediction methods valid for complex loading
conditions. Both the finite element method and a method based on Lekhnitskii’s complex stress
functions have been used to determine the stress distribution around the hole. In the case of
bolted laminates, the frictionless contact between the bolt and the hole has been taken into
account. Different failure criteria, including the Point Stress Criterion (PSC) and the Damage
Zone Criterion (DZC) have been used to predict the strength of test specimens subjected to
complex loading conditions. In the case of bolted laminates, simple failure criteria for
preliminary design are proposed. The validity of these criteria is demonstrated on multifastener tension- and shear-loaded test specimens. A comprehensive test programme was
carried out to establish criteria parameters and to generate data to validate the stress analysis
and strength prediction under complex loading conditions. A special test fixture was designed
for this purpose. Good agreement was found between predicted and measured results.
The objective of the work on three-dimensional methods has been to develop tools for threedimensional stress and failure analyses and to study the effect of various factors on the
strength of single-lap joints. An experimental programme was conducted to characterize the
failure mechanisms and to measure deformation, strain, and bending effects. A threedimensional finite element model of a bolted single-lap joint has been developed to determine
the non-uniform stress distribution through the thickness of the laminate in the vicinity of the
hole. The model was validated against measured strains and displacements from the
experimental programme. The agreement between predicted and experimental results was
generally good. In the failure characterisation part of the experimental programme, it was
found that the failure of the joints was dominated by kinking. A failure analysis procedure for
the prediction of bearing failure dominated by kinking has been developed. In this procedure,
failure is predicted using a quadratic failure criterion evaluating fibre stress and transverse
shear stress at a characteristic distance from the edge of the bolt hole. Experimental results
were used to validate the analysis procedure and good agreement was found between
predicted and experimental failure loads. The analysis procedure were then used study the
effect of eight different parameters which affect the strength of bolted composite laminates.
The parameters studied were: laminate thickness, lay-up, bolt diameter, bolt configuration
(countersunk or protruding head), friction, clamping force, clearance and lateral support. The
parametric study was organised as a reduced two-level factorial test. The results from the
parametric study show that laminate thickness, friction, clamping force and bolt configuration
are the parameters which have the strongest influence on the strength of the joints.
ii
CONTENTS
PREFACE .......................................................................................................................... i
ABSTRACT...................................................................................................................... ii
CONTENTS.....................................................................................................................iii
DISSERTATION.............................................................................................................. iv
INTRODUCTION ............................................................................................................ 1
1
Background ............................................................................................................. 1
2
Stress analysis.......................................................................................................... 3
2.1
Laminates with open holes ................................................................................. 4
2.2
Bolted laminates ................................................................................................. 5
3
Failure prediction..................................................................................................... 9
3.1
Failure prediction in laminates with open holes ................................................. 9
3.2
Failure prediction in laminates with bolted joints ............................................ 13
4
Design methods ..................................................................................................... 15
SUMMARY OF APPENDED PAPERS......................................................................... 17
DIVISION OF WORK BETWEEN THE AUTHORS................................................... 19
REFERENCES ............................................................................................................... 20
PAPER A................................................................................................................ A1-A33
PAPER B.................................................................................................................B1-B37
PAPER C.................................................................................................................C1-C46
PAPER D................................................................................................................ D1-D35
PAPER E ................................................................................................................. E1-E28
iii
DISSERTATION
This dissertation consists of an introduction and the following five papers:
PAPER A:
Ireman T., Nyman T. and Hellbom K., “On Design Methods for Bolted
Joints in Composite Aircraft Structures”, Revised version of paper with
same title, published in Composite Structures, Vol. 25, pp. 567-578, (1993).
The original paper was presented at the Seventh International Conference on
Composite Structures (ICCS/7), 9-11 July 1993 in Paisley, where it received
the Elsevier Composite Structures Award.
PAPER B:
Ireman T. and Eriksson I., “Strength of Composite Laminates Containing
Holes and Subjected to Complex Loading Conditions”, Journal of
Composite Materials, Vol. 31, pp. 1214-1248, (1997).
PAPER C:
Ireman T., Ranvik T. and Eriksson I., “On Damage Development in
Mechanically Fastened Composite Laminates”, To be submitted for
publication in Composites Part B.
PAPER D:
Ireman T., “Three-dimensional Stress Analysis of Bolted Single-lap Joints”,
Accepted for publication in Composite Structures.
PAPER E:
Ireman T. and Purin P., “Ranking of Factors Affecting the Static Strength of
Bolted Composite Laminates”, Extended version of the paper “Evaluation of
Factors Affecting the Design of Bolted Composite Laminates” which has
been accepted for presentation at the international conference “Joining and
Repair of Plastics and Composites” 16-17 March 1999, IMechE HQ,
London and publication in the Conference Transactions.
iv
INTRODUCTION
1
Background
A composite material consists of two or more materials mixed together to give a material with
good properties. A typical composite material consists of a material with high mechanical
strength and stiffness (reinforcement), for example unidirectional or woven fibres, embedded
in a material with lower mechanical strength and stiffness (matrix). To tailor the properties of
the composite material, a laminate is formed by stacking on top of each other layers of
reinforcement oriented in different directions.
Composite materials have an old history. Composites are often found in nature itself. For
example, wood is a unidirectional composite material where cellulose fibres are embedded in a
lignin matrix, and this material has been used by humans in all times in different types of
applications. The first composite laminates were made by bonding together thin plies of wood
with different fibre orientations. This type of laminate is known under the name ply-wood and
has been used in many different applications. The classical laminate theory was originally
developed for ply-wood.
Composite materials such as glass-, aramide-, boron- and carbon-fibre-reinforced plastics
have been used for a few decades, especially in the aircraft industry. At Saab, composites were
introduced in the fighter aircraft J35 Draken which was developed in the beginning of the
fifties. The Draken has parts of the air inlets made of glass-fibre-reinforced epoxy resin. The
use of composites has since then been gradually increased. On the JA 37 Viggen, the fin and
landing gear doors are honeycomb sandwich constructions with skins of carbon-reinforced
epoxy resin. The definite breakthrough for composites came with the JAS 39 Gripen in which
about 20% of the structural weight is composites. The use of composites in JAS 39 Gripen is
shown in Fig. 1. The major composite structural parts are: the wing, fin and canard.
Equipment door
Canard
Vertical fin
Fillet
Radome
Nose landing gear doors
Main landing gear doors
Fig. 1. The use of composites on the JAS 39 Gripen.
1
Leading edge
flaps
Composite materials, if properly used, offer many advantages over metals. Examples of such
advantages are: high strength and high stiffness-to-weight ratio, good fatigue strength,
corrosion resistance and low thermal expansion. Nevertheless, conventional composites made
of pre-impregnated tape or fabric also have some disadvantages, such as poor transverse
properties, inability to yield and sensitivity to moisture and high temperatures, which must be
accounted for in the design.
Among the most important elements in aircraft structures in general and in composite
structures in particular are mechanically fastened joints. Improper design of the joints may
lead to structural problems or conservative design leading indirectly to overweight structures
and high life-cycle cost of the aircraft. Typical examples of mechanically fastened joints in
composite aircraft structures are: the skin-to-spar/rib connections in e.g. a wing structure, the
wing-to-fuselage connection and the attachment of fittings etc. Examples of such joints in the
JAS 39 Gripen aircraft are shown in Fig. 2.
Skin to spar attachment
Wing to fuselage attachment
CFC-skin
CFC-skin
CFC-skin
CFC-spar
AL-fitting
Fig. 2. Examples of bolted joints in the JAS 39 Gripen.
Aircraft structures also include a large number of open holes and cut-outs e.g. holes for
electric wires and hydraulic pipes or holes required for assembly or maintenance. This is
exemplified in Fig. 3, where a typical wing spar from JAS 39 Gripen is shown. This is a typical
example of a case where a laminate containing open holes is subjected to shear loading i.e. a
2
two axial loading case. To avoid the problems due to improper design mentioned above, and to
fully utilize the material, the design methods must be applicable to general in-plane loading
cases.
Fig. 3. Typical wing spar from the JAS 39 Gripen.
This study can be divided into two parts. The first part, is focused on design methods for
laminates containing bolt holes and open holes subjected to multi-axial loading conditions.
The second part is focused on three-dimensional effects in bolted composite single-lap joints.
The objective of the first part of this study was to develop accurate and cost-efficient methods
for the stress and failure analysis of composite laminates containing bolt holes and open holes
subjected to multi-axial loading conditions. The objective of the second part of this study was
to characterize experimentally the bearing failure mode in single-lap joints, to develop tools
for three-dimensional stress and failure analyses and to study the effect of various factors on
the strength of single lap joints.
2
Stress analysis
Design methods for laminates containing bolt holes and open holes require a detailed
knowledge of the stress distribution in the vicinity the hole. This stress distribution can be
determined by analytical methods and numerical methods such as the Finite Element Method
(FEM), depending on the degree of simplification introduced. In this chapter, different aspects
on stress analysis methods for bolted laminates and laminates with open holes are discussed,
and the areas addressed in this work are pointed out.
3
2.1
Laminates with open holes
Consider a laminated composite plate of arbitrary shape containing an open hole and subjected
to general in-plane loading, as shown in Fig. 4.
90o
+45o
0o
-45o
Fig. 4. Laminated composite plate containing a hole and subjected to general in-plane
loading.
To determine fully the stress state in the plate, a three-dimensional analysis is required. A three
dimensional stress state appears at free edges such as hole edges due to the so-called “free
edge” effect caused by the mis-match of Poisson’s ratios between plies with different
orientations in a laminated plate. This gives rise to interlaminar shear and normal stresses at
the free edge. An accurate analysis of the stress state at a “free edge” requires that all the
individual plies and preferably also the resin-rich zone between the plies are represented in the
model. If the finite element method is used, a very fine mesh is needed close to the free edge in
order to capture the steep stress gradients, unless elements with a high degree of interpolation
polynomials are used. This, in combination with the model requirements in the thickness
direction, leads to large FE-models which require much computer power for their solution.
Several authors [1-7] have performed FE-analyses to determine the three-dimensional stress
state in laminates with circular holes. In most of the studies [1-4], conventional solid elements
were used. Singular elements were used in [5] and special elements with a series expansion in
the thickness direction were used in [6]. Nyman and Friberg [7] used the p-adaptive FEsystem STRIPE. With two exceptions, the work by Carlsson [4] and the work by Nyman and
Friberg [7], all studies were performed on laminates with only a few plies. The analysis
performed by Carlsson [4] was repeated in an unpublished work by Andersson [8] who also
4
used the p-adaptive FE-system STRIPE. In a p-adaptive FE-method, it is possible to increase
the degree of the interpolation polynomial, up to a high level. The results of this study differed
considerably from those in [4], indicating that FE-solutions using conventional elements, even
with very fine meshes, can be unreliable when “free edge” stresses are involved.
Analytical methods with some simplifications have also been used by some investigators [917] to determine the three-dimensional stress state close to the hole boundary. In all cases
classical laminate theory was combined with boundary layer theory in the boundary layer
region close to the hole. In two of the studies [11-12], the problem was simplified to a
fictitious straight edge problem by assuming that the curved edge can be divided into a series
of one-dimensional straight edge problems. With this simplification, it is important to account
for the interlaminar stresses caused by the in-plane stress gradient [12].
Considerable simplifications can be made if the interlaminar stresses close to the edge of the
hole are omitted, i.e. if the laminate is treated as a homogeneous anisotropic plate using
classical laminate theory. The stress analysis is performed either numerically with FEM using
membrane or shell elements or by analytical methods using a complex variable approach. FEanalyses of this kind are well established and will not be further treated here.
The method of complex functions was developed by Muskhelisvili [18] and extended to
anisotropic materials by Lekhnitskii [19] and Savin [20]. The original papers [19,20] included
exact solutions for infinite plates with circular and elliptical holes as well as approximate
solutions for oval, rectangular and triangular openings. The complex variable approach has
been used by many authors [e.g. 21-29]. In most of these studies, circular and elliptical holes
were considered. Quasi-rectangular holes were treated in [23] and [24]. Simplified
approximate solutions for infinite orthotropic plates with circular and elliptical holes were
derived in [22] and [25]. The exact solutions obtained with the complex variable approach is
limited to infinite plates. The standard procedure, well known for metals, is to use finite width
corrections to overcome this limitation. The finite width correction factors for isotropic plates
have frequently been used for orthotropic and anisotropic plates. The validity of doing this is
questionable since the finite width correction factors for isotropic plates are not in general
valid for orthotropic and anisotropic plates. Expressions for finite width correction factors for
anisotropic plates have been presented in [30].
In the present work, the stress analysis was preferably carried out in two steps. In the first step
(load distribution analysis), the far-field stress distribution in the laminate was determined, and
in the second step, a detailed stress analysis of the region containing the hole was performed in
order to determine the stress distribution around the hole. The load distribution analysis is
carried out using FEM. For the detailed stress analysis, a method based on Lekhnitskii’s
complex stress functions was developed and implemented in a computer code.
2.2
Bolted laminates
The analysis of laminates containing bolt holes is considerably more complex than the
analysis of laminates containing open holes. The stress state in the vicinity of a bolt hole
depends on many complex parameters such as friction. The most important parameters are:
5
geometry and stiffness of the joined members, joint configuration, friction properties of the
members being joined, clamping force and loading conditions. To include all these parameters
directly in a stress analysis of a joint is almost impossible, even with the most powerful
computers now available.
The analysis of a bolted joint in a complex structure such as an aircraft is preferably divided
into two or more steps as shown in Fig. 5. In the first step, the internal load distribution in the
joint is determined using one or a few successively refined analyses. The local stress
distribution in the laminate around the fastener is then determined in a detailed stress analysis.
Global structural analysis
Load distribution analysis
3
1
Local stress analysis
2
Fig. 5. Overall design procedure for bolted joints.
For a joint with complex geometry and loading conditions, FEM is the only way to determine
the load distribution. Load distribution analysis with FEM has been briefly discussed by the
present author and his co-authors [31] and by Eriksson [32]. Poon [33] brought up the subject
in his literature review on the design of mechanically fastened joints in composite structures.
A FE-based design procedure including both load distribution and detailed stress analysis has
6
been developed by Eriksson et al. [34]. It should be pointed out that the load distribution
analysis is a very important step in the design of bolted joints in composite structures. If the
load distribution analysis is of poor quality it does not matter how good the local stress
analysis and failure prediction are. Improved modelling techniques as well as detailed studies
of the joint behaviour are needed.
The determination of the local stress distribution in a bolted laminate is in general a threedimensional problem. The three-dimensional stress state is due to bending effects and to
clamping of the fastener. There are three kinds of bending effects: primary bending, secondary
bending and fastener bending (Fig. 6). Primary bending is caused by an external bending
moment acting on the joint, whereas secondary bending is caused by the eccentricity in e.g. a
single shear joint. Fastener bending occurs to some extent in every joint with a shear-loaded
fastener. The bending and tilting of the fastener give rise to a non-uniform stress distribution
through the thickness of the laminate, as illustrated in Fig. 7 where a normalized contact stress
distribution for the bolt-hole contact in a single-lap joint is shown. Because of the large
amount of modelling work and the complexity of the analysis, three-dimensional analyses of
composite bolted joints are relatively sparse. Only a few studies [35-48] have been reported.
a) Primary bending
b) Secondary bending
c) Fastener bending
Fig. 6. Bending effects in bolted joints.
7
Normalized
contact pressure, p ⁄ p 0
normalized contact pressure
3
2.5
2
1.5
1
0.5
0
360
1
270
0.75
180
0.5
90
[deg]
Angle, αangle
(degrees)
0.25
0
0
–z ⁄ t 1
normalized thickness
Fig. 7. Normalized contact stress distribution in a single-lap joint.
Two-dimensional analyses in which the laminate is treated as a homogeneous continuum have
been performed by several authors [31] and [49-70]. The local stress distribution in the
vicinity of the hole has been determined either with the complex variable approach [31, 49-57]
or numerically by FEM [35-48] and [58-70].
The contact between the bolt and the laminate has been treated in several different ways. One
simple way is to assume a contact stress distribution. The sinusoidal stress distribution
(frequently used for isotropic materials) has also been frequently used for orthotropic
materials [51,58,60-62]. If the bolt is assumed to be rigid, the prescription of zero radial
displacements at an assumed contact area of the hole boundary is another simple way of
modelling the contact [35,59,68]. Chang [69] used the same technique but added an iterative
procedure in which the extent of the contact area was determined. In [36], the hole periphery
was loaded by an arrangement of pin-jointed bars connected between the centre of the hole
and nodes at the loaded edge of the hole. The forces in the bars were then examined by an
iterative procedure. If the average of the bar forces in the thickness direction was tensile, the
bars were removed from the model. A similar approach was used in [66,67] where node pairs
on the bolt and laminate were connected by non-linear truss elements with zero tension
stiffness. In [63] and [65], a nodal point iteration technique was used to determine the contact
stress distribution. The complex variable approach has also been used to solve the contact
problem [31,49-57]. In nearly all cases the collocation method has been used, i.e.
undetermined coefficients in an assumed series expansion have been determined by imposing
the contact conditions in a number of points at the hole boundary. The extension of the contact
area is determined through an iterative procedure. A rigid bolt was assumed in all studies using
the complex variable approach, except in the work by Hyer et al. [50,55,56]. The effect of an
elastic bolt on stress distributions in the vicinity of the bolt hole was shown to be very small.
8
Friction has been included in [37,50,52-57,63-65]. The effect of multiple holes on the local
stress distributions in the vicinity of the bolt holes was considered in [53,57,60]. Infinite plates
were considered in most of the studies. Only in the studies in [51,52,57] were finite plates
considered using a boundary collocation method and a boundary integration method,
respectively.
In the first part of the present work, a stress analysis method based on the complex variable
approach was developed, the frictionless contact between the bolt and the hole being taken
into account. This method and FEM have been used to calculate the stress distributions in
three different pin-loaded laminates with two different widths. Two FE-models with different
mesh divisions were used to study the influence of the mesh density on the calculated stress
distribution. The agreement between the stress distributions obtained with the two methods is
good. The stress analysis method based on the complex variable approach is a part of a design
program for composite bolted joints used at Saab AB. In this program, the stress analysis can
be performed on anisotropic laminates subjected to bolt loads and to combined normal and
shear by-pass loads.
In the second part of the work, a three-dimensional FE model was developed to determine the
non-uniform stress distribution through the thickness of the laminate in the vicinity of the hole
in single-lap joints. To validate the model, an experimental programme was conducted in
which strains in the vicinity of the hole and relative displacements between the joint members
were measured. Computed and measured strains and displacements were compared and good
agreement was generally obtained.
3
Failure prediction
Design methods for laminates with open holes and laminates in bolted joints also require
criteria for strength prediction. In this section, strength prediction for laminates with both open
holes and bolt holes are discussed. The first part of the present work focuses on the strength
prediction of laminates containing open holes and bolt holes and subjected to general in-plane
loads. The second part focuses on the strength prediction of tension-loaded, bolted single-lap
joints where through-the-thickness effects are taken into account.
3.1
Failure prediction in laminates with open holes
Consider again the in-plane loaded laminated plate shown in Fig. 4. The load at which the
plate is unable to support any additional load is defined as the failure load. At this point,
damage due to different failure mechanisms may be present in the laminate, depending on the
stacking of the laminate. Fig. 8 shows some of the internal failure types that may occur.
9
Section B
Section A
Section A
Matrix cracks
Fibre failure
Fibre failure
Fibre-matrix failure
Fibre-matrix failure
Section B
Matrix cracks
Delamination
Fig. 8. Failure modes in a composite ply.
These failure modes are fibre failure, matrix cracking, fibre-matrix failure and delamination.
Matrix cracking usually occurs first and the crack density increases with increased loading.
When the matrix cracks are sufficiently close, a fibre-matrix failure between two matrix cracks
occurs forming a small delamination. If the primary loading is compressive, local instability
will occur when a certain level of delamination and matrix cracking is reached. If the primary
loading is tensile and the stacking is well mixed, the final failure will be controlled by fibre
failure. Delamination may also occur due to high interlaminar stresses e.g. at the free edge of
an open hole.
The existing methods for strength prediction can be divided into two main groups: laminate
failure criteria and lamina failure models. To the first group belong criteria which are based on
laminate strength data, whereas the second group contains models which are based on ply
strength data. A comprehensive review of current failure models especially with respect to the
first group is given by Awerbuch and Madhukar [71].
One of the first attempts to predict the strength of notched laminates was made by Waddops et
al. [72], who used the principles of linear fracture mechanics (LEFM) to form the Inherent
10
Flaw Model (IFM). The use of LEFM has been questioned by Kanninen et al. [73], who
pointed out that damage in a composite laminate caused by several failure modes is quite
different from a simple through-the-thickness crack in metals, and that self-similar crack
growth is not likely to occur in a composite laminate.
Strength predictions based on the maximum stress at the hole edge are in general very
conservative for composite laminates. In order to overcome the severe conservatism in using
the maximum tangential stress at the hole edge, Whitney and Nuismer [74,75] introduced the
principle of evaluating the stress level at a characteristic distance away from the hole edge.
They formulated the well-known Point Stress Criterion (PSC) and the Average Stress Criterion
(ASC). These criteria are simple to apply and are therefore attractive to designers. Both the
PSC and ASC have been widely used to predict the strength of laminates with open holes and
bolted laminates [25] and [76-80]. One problem with these models is that the characteristic
length is a function of both notch type and notch size for the same laminate. For circular holes,
this problem was addressed by Karlak [81] and Pipes et al. [82], who proposed relationships
between the characteristic distance used in PSC and the hole radius.
Bäcklund [83] concluded that IFM, PSC and ASC are all semi-empirical in the sense that the
criteria parameters can not be calculated or postulated on the basis of fundamental data for the
composite material. A new model, the Fictitious Crack Model (FCM), later renamed the
Damage Zone Model (DZM), was proposed for composite materials. The DZM simulates the
damage development in the stress-intense region at the edge of the notch by introducing a
crack with cohesive stresses acting at the crack surfaces. The cohesive stresses are assumed to
be linear functions of the crack opening. The model is based on the two fundamental
parameters, the unnotched tensile strength and the apparent fracture energy. The DZM has
been used in a number of studies to predict the strength of laminates containing cracks and
holes of various shapes [84-86]. Very good agreement between predicted and experimental
strength has been obtained in all studies. The numerical procedure of the DZM makes it less
attractive to designers than e.g. ASC and PSC.
Eriksson and Aronsson [87] developed the Damage Zone Criterion (DZC), which is based on
the two fundamental laminate parameters unnotched tensile strength and critical damage zone
length. The failure load is determined from a simple equilibrium equation at the point of
failure when the damage zone has reached a critical size. The stresses within the damage zone
are assumed to be constant and equal to the unnotched strength of the laminate, and the stress
distribution ahead of the damage zone is assumed to have the same shape as the linear elastic
stress distribution.
The lamina failure criteria can be divided into two groups: first-ply failure and progressive
failure criteria. The first group contains characteristic distance approaches applied at the plylevel [89-95]. Garbo and Ogonowski [89] evaluated the Tsai-Hill failure criterion at a
characteristic distance from the hole while El-Zein and Reifsnider [90] used the principle of
the Average Stress Criterion (ASC) at the ply level by assuming that failure occurs when the
average ply fibre stress over a characteristic distance is equal to the critical fibre stress. Tan
[91-95] evaluated a fibre failure criterion and the Tsai-Wu criterion along a characteristic
curve.
11
In the progressive failure criteria, the stiffness properties of the material are successively
degraded as the plies fail. The failure of a ply is predicted using e.g. Tsai-Wu, Tsai-Hill,
Hashin etc. and the material degradation is accomplished by setting some of the components
in the lamina stiffness matrix to zero depending on the type of failure. Examples of such
damage models are found in [96-100]. Delamination was included in some of the models
[96,98,99] by using stress-based polynomial criteria in terms of the interlaminar shear and
peel stresses.
Predictions of delamination onset at the free edge of holes in composite laminates are made in
[5] and [12-16]. Ericson et al. [5] used a fracture mechanics approach evaluating the total
strain energy release. Shalev and Reifsnider [13] used a criterion based on the order of
singularity of the stress field and the energy release rate. Stress-based polynomial criteria in
terms of the interlaminar stresses evaluated at a characteristic distance from the hole edge
were used in [12] and [14-15].
In most of the studies of laminates with holes, uniaxial loading conditions have been
considered. Only a few authors [21,94,101-104] have considered laminates with holes
subjected to more complex loading situations such as biaxial and shear loading. Whitside et al.
[21] included a biaxial tension-tension test in their comprehensive experimental study on
different types of fibre-reinforced epoxy laminates. A similar test procedure was used by
Liebowitz and Jones [101] and by Daniel [102,103] who performed biaxial tension-tension
tests with different ratios between the loads in the two loading directions. Tan [94] performed
shear tests on laminates with elliptical holes using a rail shear test fixture. Gamziukas and
Carlsson [104] performed shear tests on I-beams with an irregular cutout.
Studies on the strength prediction of notched laminates subjected to general in-plane loading
situations such as biaxial loading are even more sparse than the experimental studies on the
subject. Lee [96] analysed the biaxial tests in [101] using a progressive failure criterion. Tan
[94] analysed shear tests as well as the biaxial test by Daniel [102,103] using a ply-failure
approach applied at a characteristic distance from the hole edge. Hollman [88] analysed the
shear-loaded graphite/epoxy beam with an irregular cut-out in [104], using the Point Stress
Criterion (PSC) and the Damage Zone Model (DZM) for strength prediction.
The major advantage of the laminate strength prediction methods over the lamina strength
prediction methods is that the complex mechanisms of laminate failure including fibre failure,
matrix cracks and delamination can be taken into account in a simple and accurate way. A
general limitation of these methods is that tests have to be made on each laminate
configuration used in the design in order to determine the laminate fracture parameters. This
limitation can, however, be effectively overcome by establishing analytical and experimental
models to determine the fracture parameters using only a limited amount of test data.
The lamina models on the other hand have the advantage that the strength of the laminate is
predicted from basic strength parameters for the ply, i.e. the amount of testing required to
predict the strength for a wide range of laminate configurations is generally less than for the
laminate models. It is, however, very difficult to find a failure criterion which is capable of
predicting the strength of a laminate containing several fibre orientations using strength
12
parameters determined from tests with unidirectional laminates. Another disadvantage of the
lamina progressive failure models is that the analysis is expensive and time-consuming. A
more dense finite element mesh is generally required for the lamina progressive failure models
than for the laminate models for accurate predictions.
In this work, a series of experiments with both uniaxially loaded laminates and laminates
subjected to combined tension and shear loading was carried out. The strengths of these
specimens were predicted using the PSC and the DZC extended to general in-plane loading
situations. Good agreement between predicted and experimental strengths is obtained with
both criteria.
3.2
Failure predictions in laminates with bolted joints
A bolted joint in a composite structure can fail in a number of ways as shown in Fig. 9. Bolt
failure is often a secondary failure mode in composite structures i.e. the bolt fails after the
laminate has failed in bearing. Pull-through failure is the predominant failure mode in joints
with axially loaded fasteners or a secondary failure mode in a single shear joint with shear
loaded fasteners. This failure mode will not be treated here. The remaining failure modes:
bearing, net-section and shear-out are the basic failure modes of a laminate in a composite
bolted joint with shear-loaded fasteners. The shear-out failure mode occurs mainly in joints
where the distance between the hole edge and the edge of the laminate is short or in highly
orthotropic laminates such as cross-ply laminates. The shear-out failure mode can be avoided
by using appropriate design rules for edge distances and stacking sequences. Shear-out failure
will not therefore be further addressed here.
a) Net-section failure
b) Bearing failure
d) Fastener failure
Fig. 9. Failure modes for composite bolted joints.
13
c) Shear-out failure
Net-section failure in tension or compression is assumed to occur by the same mechanisms as
for laminates with open holes. The same failure criteria as for open holes which were
described in the previous section, are therefore used to predict net-section failure of laminates
in bolted joints.
Bearing failure is primarily a compressive failure occurring close to the contact region at the
hole edge. The failure is caused by the compressive contact stresses acting on the hole
boundary. Bearing failure is, like any compressive failure in a composite laminate, associated
with delamination and ply-buckling, which implies that the bearing strength is strongly
affected by the lateral constraint of the material surrounding the loaded hole. The effect of the
lateral constraints on the bearing strength has been shown experimentally in [105-109].
Eriksson [112] formulated a bearing strength prediction model, the Delamination Buckling
Model (DBM), from stability considerations. The bearing strength is also strongly dependent
on the layup of the laminate [110] and is sensitive to environmental conditions such as high
temperature and high moisture content in the laminate [111].
With the exception of the DBM, the strength prediction methods for bearing failure are either
of the laminate type [51,59] or the lamina type [58,61,66-68,113], according to the description
in the previous section. At the laminate level, the ASC [51,59] in terms of the radial stress has
been used to predict the bearing strength with characteristic distances determined from
bearing strength tests. In the strength predictions studies with lamina methods, various ply
failure criteria such as Distortional Energy, Tsai-Wu, Yamada-Sun have been used either in
combination with a characteristic distance or as progressive failure models.
Relatively few authors deal with the failure analysis of a bolted joint on the basis of a threedimensional stress state [39,42,46,47]. Chen and Lee [39] used the maximum stress theory to
predict in-plane fibre, matrix and shear failure and the Ye delamination criterion [41] in a
progressive damage model. A progressive fatigue modelling technique which includes a
progressive failure analysis has been developed by Shokrieh et al. [45]. A set of stress-based
failure criteria were used to predict matrix tension, matrix compression, fibre tension, fibre
compression, fibre-matrix shear out, delamination tension and delamination compression.
Gamble et al. [42] used a modified version of the Hill-criterion in a progressive damage model.
The criterion was able to predict fibre failure, matrix splitting and delamination. Delamination
initiation in laminates with pin-loaded holes was predicted by Persson et al. [47] using the strain
energy density criterion.
In the first part of this work, two very simple stress-based criteria were proposed for the
prediction of tension and bearing failure in composite bolted joints, and the validity of the
criteria has been demonstrated in tests with realistic tension- and shear-loaded multi-fastener
joints. Comparisons between experiments and predictions show that the strength of these
joints is fairly well predicted with the simple failure criteria.
In the second part, an experimental programme was conducted to characterize the damage
development in single-lap joints. Based on the results from this experimental investigation, a
failure analysis procedure for the prediction of bearing failure dominated by kinking was
developed. The failure analysis procedure was validated against experiments and good
14
agreement between predicted and experimental results was generally found. The failure
analysis procedure was then used together with three-dimensional FE analyses to study
numerically the effect of different factors on the strength.
4
Design methods
Several analysis programs for composite bolted joints are available for composite structure
designers. Snyder et al. [114] examined six different analytical programs and discussed their
merits and disadvantages. In addition to these programs, Madenci [115] has developed a
design and analysis program which is capable of handling multi-fastener joints with
interaction between the fasteners, finite geometry, friction and by-pass loading. All the
programs are analytically based and most of them use the complex variable approach. There is
only one program among those examined in [114] and the program developed by Madenci in
which the contact stress distribution between the bolt and the hole is determined. There are,
however, a couple of FE-based programs in which the contact stress distribution is determined,
[32] and [34,116].
The stress analysis method based on the complex variable approach used in this work for the
analysis of laminates in bolted joints and laminates with open holes has been used as a basis
for a design program for bolted joints called COBOLT. COBOLT is capable of analysing a
large number of joints using both simple and more advanced failure criteria. It is possible
either to solve the frictionless contact problem or to use a fixed cosine distribution in the stress
analysis.
All the design and analysis codes already mentioned are two-dimensional whereas the stress
state in a bolted joint is in general three-dimensional. To account for the three-dimensional
effects, the two-dimensional analyses have to be corrected by correction factors. A design tool
based on a three-dimensional analysis would make it possible to include the bending effects in
the local stress analysis and thereby reduce the conservatism in the design. As a first step
towards such a design tool, a mesh generation programme which generates three-dimensional
FE models of an isolated region around a fastener (Fig. 10) has been developed. This tool has
partly been used for the generation of three-dimensional FE models in this work.
15
3
1
2
Fig. 10. Three-dimensional FE model of an isolated region around a fastener.
16
SUMMARY OF APPENDED PAPERS
PAPER A
The problems related to the determination of the load distribution in a multi-row fastener joint
using the finite element method are discussed. Both simple and more advanced design methods
used at Saab Military Aircraft are presented. The stress distributions obtained with an
analytically based method and an FE-based method are compared. Failure predictions with a
simple analytically based method and the more advanced FE-based method of multi-fastener
tension and shear-loaded test specimens are compared with the results of experiments. Finally,
complicating factors such as three-dimensional effects caused by secondary bending and
fastener bending are discussed and suggestions for future research are given.
PAPER B
Stress and failure analyses of composite laminates containing holes are presented. Both the
finite element method (FEM) and an analytical method, based on complex stress functions, are
used to determine the stress distributions around the holes. Failure criteria, including the Point
Stress Criterion (PSC) and the Damage Zone Criterion (DZC), are used to predict the strength
of test specimens subjected to complex loading conditions. A comprehensive test program was
carried out to establish criteria parameters and to generate data to validate the stress and failure
analysis. For this purpose, a special test fixture was designed. Good agreement was found
between predicted and measured results.
PAPER C
A comprehensive experimental program was conducted to measure and characterize the
development of damage in the vicinity of fastener holes in graphite/epoxy composite laminates.
This was carried out to generate data which can be used for development of appropriate failure
criteria. Test specimens were loaded in quasi-static cycles with successively increasing loads,
and damage development in the vicinity of the bolt holes was detected using different methods
such as strain measurements, acoustic emission, X-ray and microscopic examination. Several
failure modes were detected in a series of events starting at load levels far below the level at
which the first visible evidence of damage appeared on the load-displacement curve. Failure
modes included matrix cracking, fibre fracture, delamination, and kinking.
PAPER D
A three-dimensional finite element model of bolted composite joints has been developed to
determine non-uniform stress distributions through the thickness of composite laminates in the
vicinity of a bolt hole. An experimental programme was conducted to measure deformation,
strain, and bolt load on test specimens in order to validate the numerical model developed.
Strains in the radial direction at different radii and different angles were measured in the
vicinity of the bolt hole at the shear plane between the plates. The degree of secondary bending
17
in the joints was determined from strain measurements at certain points on both sides of the
laminate. In the experiments, a number of parameters such as laminate lay-up, laminate
thickness, bolt diameter, bolt type, clamping force and lateral support were varied. Each
specimen configuration was analysed using the three-dimensional finite element model. In
general, the computed and experimental results showed good agreement.
PAPER E
A systematic parametric study has been carried out to determine the effect of eight different
parameters which affect the strength of bolted composite laminates. The parameters studied
were: laminate thickness, lay-up, bolt diameter, bolt configuration (countersunk or protruding
head), friction, clamping force, clearance and lateral support. The parametric study has been
organised as a reduced two level factorial test. Finite element analyses were used to simulate
the experiments. Three-dimensional finite element analyses were used to determine the stress
distributions in the vicinity of the bolt hole. Failure was predicted using a quadratic failure
criterion evaluating fibre stress and transverse shear stress at a characteristic distance from the
edge of the bolt hole. Experimental results were used to validate the analysis procedure. Good
agreement between predicted and experimental failure loads was found. The stress and failure
analyses were then used in the numerical parametric study. The results from the parametric
study show that laminate thickness, friction, clamping force and bolt configuration are the
parameters with the strongest influence on the strength of the joints.
18
DIVISION OF WORK BETWEEN THE AUTHORS
PAPER A:
Ireman and Nyman developed the analytical method. Ireman implemented the method into a
computer code and carried out all the analyses. Hellbom contributed with the experimental
results. Ireman wrote the paper and initiated the work.
PAPER B:
Ireman and Erikson extended DZC to two axial loading conditions. Ireman initiated the work,
planned the experiments, implemented the method, carried out all the analyses and wrote the
paper.
PAPER C:
Ireman initiated the work and wrote the overall test plan. Ranvik carried out the experiments.
All authors contributed to the writing of the paper.
PAPER E:
Ireman developed the failure criterion. Purin created the FE models and carried out some of
the analyses. Ireman initiated the work and wrote the paper.
19
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