Multi-Fastener Single-Lap Joints in Composite Structures

Multi-Fastener Single-lap Joints in Composite Structures

Johan Ekh Department of Aeronautical and Vehicle Engineering Royal Institute of Technology SE–100 44 Stockholm, Sweden

Doctoral Thesis in Lightweight Structures TRITA AVE: 2006-22 ISSN 1651-7660 ISBN 91-7178-396-2

Preface

The work presented in this doctoral thesis has been carried out between January 2000 and May 2006. Work between January 2000 and December 2003 was conducted at the Royal Institute of Technology, Department of Aeronautical and Vehicle En- gineering and the remaining part was performed on part time basis at ABB AB, Corporate Research. A major part of the work was conducted within the framework of BOJCAS - BOlted Joints in Composite Aircraft Structures, a RTD project par- tially funded by the European Union under the European Commission GROWTH programme, ’New Perspectives in Aeronautics’. All experiments were performed at The Swedish Defence Research Agency (FOI). The support from these organizations is gratefully acknowledged. Firstly, I would like to thank my co-author Dr. Joakim Schön for his consistent support and encouragement. His knowledge and experience in the field of composite bolted joints have been invaluable to my research. I also want to thank Professor Dan Zenkert for accepting the role as my supervisor during the second part of my work. Professor Jan Bäcklund is acknowledged for accepting me as a graduate student. Finally, I thank my fiancéAnna-Karin and our children, Oskar, Matilda and Sofia, for their support and patience. This work would not have been completed without their support.

V¨aster˚as, May 2006

Johan Ekh

Abstract

This thesis deals with composite joints. Designing such joints is more difficult than metallic joints due to the mechanical properties of composite materials. Composites are anisotropic and have a limited ability of yielding. The low degree of yielding means that stress concentrations are not relieved by plastic deformation, which is important in multi-fastener single-lap joints. The distribution of load between the fasteners may be more uneven than in metallic joints due to that the stress concentrations around the holes are not relieved. Single-lap joints have an eccentric load path which generates a nonuniform bolt-hole contact pressure through the plate thickness. This generates out-of-plane deflection of the joint, termed secondary bending. Such nonuniform contact stress severely limits the strength of the joint. The nonuniform contact stress distribution is affected by several factors, e.g. bolt- hole clearance and secondary bending. The first part of the work is devoted to investigating secondary bending, and its effect on stresses in the joint. A novel technique to study secondary bending has been developed and used in a parametric study. It is based on the calculation of specimen curvature from out-of-plane deflections measured with an optical technique. It is shown that the specimen curvature is correlated to the conventional definition of secondary bending, which involves strain measurements on both sides of the plate. The two most important parameters affecting specimen curvature was found to be the overlap length and the thickness of the plates. The finite element method was used to study the influence of secondary bending on joint strength. Secondary bending was changed in magnitude by altering the length of the overlap region in a two-fastener specimen. It was found that secondary bending affects the local stress field around the fasteners and that it may change the strength and the mode of failure. The second part is concerned with the load distribution and prediction of joint strength. A detailed finite element model was developed to calculate the load distribution while accounting for bolt-hole clearances, bolt clamp-up, secondary bending and friction. An experimental programme was conducted in order to validate the finite element model by means of instrumented fasteners. Good agreement between simulations and experiments was achieved and it was found that bolt-hole clearance is the most important factor in terms of load distribution between the fasteners. Sen- sitivity to this parameter was found to be large, implying that temperature changes could affect the load distribution if member plates with different thermal expansion properties are used. Calculating the load distribution in structures with a large number of fasteners is in general not feasible with detailed finite element models based on continuum elements. A simplified, computationally effective model of a multi-fastener, single- lap joint has been developed by means of structural finite elements. The model accounts for bolt-hole clearances, bolt clamp-up, secondary bending and friction. Comparisons with the detailed finite element model and experiments validated the accuracy of the simplified model. A parametric study was conducted where it was found that an increased stiffness mismatch between the plates generates a more uneven load distribution, while reducing the length of the overlap region has the opposite effect. Increasing the stiffness of a fastener shifts some of the load from

iii the nearest fasteners to that particular fastener. An idealized optimization study was conducted in order to minimize bearing stresses in the joint with restrictions on the increase of joint weight and net-section stresses. Maximum bearing stress was reduced from 220 MPa to 120 MPa while both weight and net-section stresses decreased. A procedure to predict bearing strength based on the results from the simplified model was developed. It was established by an experimental programme that fiber micro-buckling is the initial failure mode. The stress state in the laminate was determined through force and moment equilibrium, based on output from the finite element model. An existing criterion was used to predict the fiber micro- buckling, and thus the initial failure. Predictions were compared with experiments which validated the method. The small computational cost required by the procedure suggests that the method is applicable on large scale structures and suitable to use in conjunction with iterative schemes such as optimization and statistical investigations.

iv Dissertation

This doctoral thesis contains an introduction to the research area and the following appended papers:

Paper A J. Ekh, J. Sch¨on, L. G. Melin, Secondary Bending in Multi Fastener, Composite-to- Aluminium Single Shear Lap Joints, Composites Part B, 36, (2005), 195-208.

Paper B J. Ekh, J. Sch¨on, Eﬀect of Secondary Bending on Strength Prediction of Composite, Single Shear Lap Joints, Composites Science and Technology, 65, (2005), 953-965.

Paper-C J. Ekh, J. Sch¨on, Load Transfer in Multirow, Single Shear, Composite-to-Aluminium Lap Joints, Composites Science and Technology, 66, (2006), 875-885.

Paper D J. Ekh, J. Sch¨on, Finite Element Modeling and optimization of Load Transfer in Multi-Fastener Joints using Structural Elements, Accepted for publication in Com- posite Structures.

Paper E J. Ekh, D. Zenkert, J. Sch¨on, Prediction of Bearing Failure in Composite Single-lap Joints, To be submitted for publication in Journal of Composite Materials.

Division of work between authors

Paper A Ekh and Sch¨on planned the experimental programme. Ekh and Melin conducted the experiments. Ekh conducted the FE-analyses. Ekh and Sch¨on wrote the paper.

Paper B Ekh conducted the FE-analyses. Ekh and Sch¨on wrote the paper.

Paper C Ekh and Sch¨on planned the experimental programme. Ekh conducted the experiments and the FE-analyses. Ekh wrote the paper with help from Sch¨on.

Paper D Ekh and Sch¨on planned the experimental programme. Ekh conducted the experiments and the FE-analyses. Ekh wrote the paper with help from Sch¨on.

Paper E Ekh and Sch¨on planned the experimental programme and Ekh conducted the experiments. Ekh and Zenkert outlined the numerical and analytical procedure. Ekh implemented the numerical routine and performed the analyses. Ekh wrote the paper with help from Zenkert and Sch¨on.

vii

Contents

Preface i

Abstract iv

Dissertation v

Division of work between authors vii

1. Introduction 1 1.1. Background ...... 1 1.2. Failureincompositejoints ...... 4 1.3. Stressanalysis...... 8 1.4. Strengthprediction ...... 15 1.5. Futurework...... 22

References 28

Paper A A1–A22

Paper B B1–B21

Paper C C1–C19

Paper D D1–D21

Paper E E1–E21

1. Introduction

1.1. Background

Composite materials are used extensively in aircraft structures. The main driver is their weight-saving potential, but composites also beneﬁt from good fatigue properties and corrosion resistance. A clear trend is that composites are used increasingly in heavily loaded primary structures. Each new generation of aircraft tends to use thicker laminates carrying more load. The use of composites in modern aircraft is ex- empliﬁed by a schematic over the new commercial airliner Airbus A380 in Figure 1.

Figure 1: Major composite parts on the Airbus A380.

More than 20% of the structural weight are composites. Typical examples of the increased use of composites in heavily loaded structures are the centre wingbox and a part of the vertical tail plane, which are illustrated in Figure 2. The wingbox weighs a total 8.8 tonnes, of which 5.3 tonnes are composite materials, and utilizes composite laminates as thick as 45 mm. A composite material is built-up from two or more constituent materials that have different properties. Usually, a discrete component with high stiffness and strength, termed reinforcement, is embedded in a continuous material termed matrix. Properties of the composite are determined by the properties of the constituents, their organization and the bonding between them. This implies that composites can be designed to have specific properties, to meet the requirements from different applications, by changing the above parameters.

1 (a) Centre wingbox. (b) Part of vertical tail plane.

Figure 2: Composite parts of the Airbus A380.

Several kinds of composite materials are being used. A distinction can be made based on the geometry of the reinforcing constituent that can be either a particle or a fiber. Fibrous composites are generally stronger and stiffer than particulate composites and therefor frequently used in demanding applications. The fibers may be short or long relative to other characteristic dimensions which makes the composite discontinuous-fiber-reinforced or continuous-fiber-reinforced. The latter kind of composite is fabricated by organizing fibers into a thin sheet of parallel fibers and saturate them with a resin. The resulting tape of pre-impregnated unidirectional fibers is called a pre-preg and is a fundamental building block in composite aircraft materials. Composite laminates are formed by stacking the pre-pregs with different fiber orientations, according to a specific stacking sequence or lay-up, and cure them in an autoclave under elevated pressure and temperature. Lightweight, strong and stiff materials suitable for aircraft design can be developed by this procedure. A fundamental feature of composites is that they are anisotropic, i.e. material properties are different along different directions. This applies to both elastic and inelastic response and is different from metals which are essentially isotropic. Aircraft materials are often compared in terms of their specific properties, i.e. the strength- to-weight and stiffness-to-weight ratios. In directions with many parallel fibers, the specific properties of composites can be superior to those of metals. However, while in-plane strength can be excellent, out-of-plane strength is usually relatively poor which must be accounted for when designing composite structures. Designing useful structures means that several parts must be joined together. This can be done by adhesive joining, mechanical joining or hybrid joining which is a combination of the first two techniques. The use of adhesive joints in heavily loaded structures is often restricted by the low out-of-plane strength of the composite. A qualitative distribution of transverse and shear stress for a typical joint is shown

2 in Figure 3. It can be seen that stress concentrations occurs at the ends of the overlapping region, thus severe stress components are generated that coincides with the weakest modes of the composite. Another disadvantage of adhesive joining is that the structure can not easily be disassembled for replacing damaged parts.

Figure 3: Interlaminar stresses in an adhesive joint.

Mechanical joining, i.e. using bolts or rivets, is the most important method in the aerospace industry. Although it is the preferred joining technique in many cases, it is still associated with difficulties. The presence of a hole in a laminated plate subjected to external loading introduces a disturbance in the stress field. Stress concentrations are generated in the vicinity of the hole. Inserting a fastener into the hole and reacting load through the contact between the fastener and the hole surface will make the stress concentrations even more severe. Because of this, bolted joints are weak spots in a composite structure and must be properly designed to achieve an efficient structure. This has prompted a significant research effort during that last few decades, aiming at developing reliable design methods. Such design methods are imperative in order to achieve safe structures with optimum performance. Conducted research is either theoretical, experimental or a combination. Theoretical work usually aims at developing models that can be used to predict joint strength. Experimental investigations have been undertaken to clarify failure behavior of joints, or to validate theoretical models. Comprehensive reviews of the field are given in [1–6]. The objective of the present work is twofold: First, a detailed investigation on out-of-plane deflection of multi-fastener single-lap joints was conducted. This involves development of a new measurement technique based on an optical full field method. The implications of out-of-plane deflection on joint strength was investigated. Second, a simplified technique to predict load transfer and bearing strength in multi-fastener composite joints was developed. The method was validated by means of an experimental programme. The outline of this introductory part of the thesis is as follows: Some fundamental aspects of joint strength are briefly discussed in Section 1.2. This forms a background

3 to Sections 1.3-1.4 where stresses and failure predictions are discussed, respectively. Finally, some possible future extensions to the work presented in this thesis are outlined in Section 1.5.

1.2. Failure in composite joints

Usage of composite materials in aircraft requires detailed knowledge about their behavior. In particular, it is important to know the strength of the joints and the mode in which they fail. Fundamental investigations on joint strength has been conducted using simple test objects. Three basic types of test specimens are illustrated in Figure 4.

(a) Pin-loaded plate. (b) Double-lap joint. (c) Single-lap joint.

Figure 4: Test specimens.

The most common specimen is the pin-loaded plate. It is characterized by the symmetry with respect to the center of the laminate and by the absence of lateral clamping of the laminate. Sometimes, however, lateral clamping of the composite is provided by using a fastener instead of a pin. The two splice plates are not considered to be part of the joint but rather of the fixture, and are usually very stiff compared to the composite middle plate. The double-lap joint differs from the pin-loaded plate mainly by the presence of lateral clamping provided by the fastener. Splice plates are considered to be part of the joint and their total strength and stiffness usually compares to those of the middle plate. Although true symmetry does not exist, as the bolt is not symmetric with respect to the center of the middle plate, it is often a good approximation which is widely used in numerical work. Single-lap joints are not symmetric with respect to the center of the joint. This nonsymmetry causes the fastener to tilt in the hole, which implies that the bolt- hole contact pressure becomes nonuniform through the plate thickness. A stress

4 concentration appears at the hole edge close to the shear surface of the joint. The nonuniform forces causes the plate to bend out-of-plane which is referred to as secondary bending (SB). Different aspects related to SB are treated in Papers A-B in this thesis. A typical test procedure is to clamp both ends of the specimen in an hydraulic test machine and apply tensional displacement to one end while keeping the other end fixed. The rate of the displacement is constant at a low value and the force is monitored continously. The resulting load-displacement curve is usually nonlinear for small loads. This is due to that the contact surface between the pin and the hole edge is growing as the load increases. At some small load, the contact surface is fully developed and the load-displacement is close to linear. A small weakening effect can usually be noted, which can be attributed to nonlinear elastic properties, or to the development of small micro-mechanical damages in the material. At some high load the joint will fail. This can either be in an abrupt mode, which means that the joint immediately becomes unable to carry load, or in a ductile mode. If ductile failure occurs, the load-displacement curve will become increasingly nonlinear. However, the joint may continue to carry an increasing load before final failure. It has been found that composite joints fail in a number of different macroscopic modes. The most frequent ones are illustrated in Figure 5. The net-section and cleavage modes are abrupt, with a well defined failure load, whereas bearing and shear-out usually are more ductile.

F F F F

(a) Net-section (b) Bearing (c) Shear-out (d) Cleavage

Figure 5: Macroscopic failure modes of composite joints.

In the case of ductile failure it becomes important to determine what defines failure of the joint. Failure load can be defined as the maximum load sustained by the joint, or as the load at which a certain hole deformation has developed. The 2% offset bearing strength criterion is outlined in ASTM Standard D5961/D5961 M- 01 [7] and means that the joint has failed when inelastic deformation corresponding to a certain hole elongation has occurred, as indicated in Figure 6(a).

Bearing stress, σb, and bearing strain, ǫb, are defined according to Equations 1 and 2, F σ = (1) b kDt 5 (δ1 + δ2)/2 ǫ = (2) b KD respectively, where F =load, D=diameter, t=laminate thickness, k=number of fasteners, K =1.0 for double-lap specimens and 2.0 for single-lap specimens and δ1, δ2 are the displacements as measured with two extensometers over the overlap region of the joint. However, different joints may be in quite different state of damage when the criterion is satisfied. The criterion may also be difficult to evaluate as it is based on the existence of a linear portion of the bearing stress-strain curve as shown in Figure 6(a). Such a linear portion must be approximated and thus involves some judgment of the operator. Recently, McCarthy et. al. [8] suggested that failure has occurred at a certain amount of reduced stiffness of the joint, as illustrated in Figure 6(b). This definition eliminates the potential cause of error in identifying the linear portion of the load- displacement of the curve. Note that the highly nonlinear behavior for small strains is not related to damages and should be disregarded.

110 100 2% Ultimate Strength 600 6 90 80 Offset Strength 70 400 4 60 50 40 Bearing Stress (ksi) Offset Line 200 2 30 Bearing Stress [MPa] Bearing Stiffness [GPa] 20 10 Stress Stiffness 0 0 0 0 5 10 15 20 25 0 2 4 6 8 10 12 14 Bearing Strain, % Bearing Strain, % (a) Example of bearing stress-strain curve. (b) Example of bearing stress-strain and stiﬀness-strain curves.

Figure 6: Experimental failure criteria.

An other definition that has been used is that failure has occurred at the first peak, or the first irregularity, of the load-displacement curve. The failure load and the transition between the failure modes are affected by several parameters, such as laminate stacking sequence, the geometry of the joint and properties of the fastener. The effect of laminate stacking sequence has been investigated by several authors [9–11]. A general trend is that quasi-isotropic, or near quasi-isotropic, laminates provide higher failure strengths and that the failure mode is either bearing or net-section in such a laminate. Reducing the number of angle plies, or transverse plies, in favor of longitudinal plies increases the risk of pre-mature failure in the shear-out mode. This implies that the stacking sequence rarely deviates much from the quasi-isotropic case. The influence of geometrical parameters, as defined in Figure 7(a), has been investigated [9, 11]. It was found that the width to diameter ratio (w/d) and the edge distance to diameter ratio (e/d) influenced the failure. Going from a small value on (w/d) to a larger value shifts the failure mode from net-section to bearing for quasi-isotropic laminates. Maximum joint strength is achieved around the transition

6 from net-section to bearing failure which seems to occur for 2.5 < w/d < 4 for many joints. Increasing (e/d) up to 4 shifts the failure mode from shear out to bearing and increases the failure load. Further increase of (e/d) has limited eﬀect.

x θd e y z

(a) Plate geometry. (b) Protruding head and countersunk fasteners.

Figure 7: Geometry of member plate and fasteners.

Fasteners can be either countersunk or protruding head and they can be finger tightened or torqued. Examples of fasteners are illustrated in Figure 7(b). Counter- sunk fasteners provide less bearing strength than protruding head fasteners due to the reduced length of the cylindrical bolt shank. The presence of lateral constraint provided by the washers stabilizes the damaged material close to the bolt, which increases the bearing strength [10,12–15]. Relatively few investigations have been conducted on composite, single-lap joints. Experimental investigations in which the influence of geometry [16–19], stacking sequence [8, 17–19], torque level [8, 16, 17], and fastener type [8, 16, 17] have been conducted. A clear trend is that single-lap joints have a reduced bearing strength compared to double-lap joints. The strength of multi-fastener joints has been investigated by several workers [19–24]. It was found that failure at a particular bolt was depending on the load reacted by the bolt and the amount of load passing by the bolt. This is referred to as the bearing-bypass ratio of the fastener and affects the failure as illustrated in Figure 8. Bearing strength in tension loading is relatively insensitive to increased bypass stress. At some point failure is shifted from bearing to net-section. Net- section is influenced by the amount of bearing stress such that the strength increases linearly with decreasing bearing stress. Under compressive loading, bearing strength is depending on the amount of bypass stress. Bypass stress tends to decrease the contact area which reduces the bearing strength. At some point, the failure mode is shifted from bearing to net-section. This has been reported to be a gradual process. Important results from these experimental investigations are:

1. Laminate stacking sequence should not deviate much from the quasi-isotropic case in bolted laminates.

7 Bearing stress

Bearing

Bearing Net-Section

Compressive Tension Load Load Net-Section

0 Bypass stress

Figure 8: Bearing-bypass interaction.

2. The most likely failure modes are the ductile bearing and the brittle net-section modes.

3. Single-lap joints have a reduced bearing strength due to the stress concentration generated by the nonsymmetric geometry. Using composites in aircraft requires proper design rules. This has motivated a large research eﬀort to establish predictive capability of joints strength.

1.3. Stress analysis

A starting point for strength analysis is to determine the level of stress in the material. The purpose of this section is to describe how stresses are developed in composite joints, and introduce some existing methods to predict these stresses. Since the number of factors affecting the stress state is large, the approach is to start with a laminated plate and gradually increase the complexity, up to general multi-fastener joints. First a note regarding the concept of stress. Embedding stiff fibers in a resin means that the material is heterogenous. This implies that the stress field is complicated and may change abruptly from point to point. Large stress concentrations and gradients can be expected at the interfaces between the fibers and the matrix. It is not practical to account for the stress field on the length scale of individual fibers when evaluating the performance of a composite structure. Instead, it is custom to make a homogenization of the material, based on the macroscopic properties of the laminate. Thus, it is assumed that either each unidirectional lamina, or the entire composite laminate, is a homogenous anisotropic material with the appropriate elastic constants.

Laminated plate Consider a composite plate of finite dimensions as illustrated in Figure 9(a). A longitudinal stress, σx, is applied on the ends of the plate. Due to the difference in stiffness and Poisons ratio between the laminae, shear stresses are developed on the

8 lamina faces. Theses stresses are small and may usually be neglected some distance away from the laminate boundaries. Free boundaries must be traction free which implies that some shear stresses must vanish as the boundary is approached. Other stress components must grow in order to maintain equilibrium in the plate. This is illustrated in Figure 9(b). In regions away from boundaries, lamina stresses can be calculated with classical lamination theory (CLT). This is a linear elastic solution under the assumption that the stacked laminae are rigidly connected to each other.

(a) Laminated plate in uniform stress. (b) Interlaminar stresses.

Figure 9: Interlaminar stresses in a symmetric laminated plate [25].

The elastic response of the matrix material can be nonlinear. This generates a nonlinear behavior of the lamina, mainly in shear deformation. In applications where individual laminae are subjected to extensive shear deformation, this may have an aﬀect on the behavior of the laminate. A linear elastic solution for the interlaminar stresses close to the boundaries was developed by Pipes and Pagano [25]. It was also shown [26] that the interlaminar stresses were strongly dependent on the laminate stacking sequence. The three-dimensional ﬁnite element method was used to study the same problem in [27].

Plate with an open hole If an open hole is introduced at the center of the plate, the stress state will change drastically. Stress concentrations will occur at the hole surface in the net-section plane. In isotropic materials the stress at the hole surface is known to be 3 times larger than the average stress. In anisotropic materials, the stress concentration can be even greater. The interlaminar stresses at the free boundaries discussed above can be expected also at the boundary of the hole. The 3D stress ﬁeld, including interlaminar stresses, has been investigated by several authors, either by the ﬁnite element method [28–33] or analytical methods [34–36].

9 The large stress gradients at free edges requires special attention when numerical methods are used. A very fine mesh at the boundary between two plies were used by Hu et. al. [33]. Jinping and Huizu [30] developed a special finite element that represented the solution analytically. This element was used close to free edges were the regular finite elements were inadequate. Regular elements were used far away from the boundaries. A substructure technique was used by Kim and Hong [31]. High order elements were used by Nyman and Friberg [37] while an approach based on spline approximations was used by Iarve [38]. The problem is simplified if the interlaminar stresses are excluded from the analysis. Several workers have used the method of complex functions [39] for anisotropic plates [40] to calculate the stress field around the hole.

Pin-loaded plate

If an elastic pin inserted in the hole is used to balance the applied stress, σx, the problem is transformed into a nonlinear contact problem. Stresses generated from contact of two conforming bodies, such as the pin and the plate, are highly dependent on the geometry of the bodies and their surface properties. An oversized hole results in a small initial contact area that will grow as the load is increased due to the deformation of the bodies. Radial contact stress and shear stresses, in both the tangential and the thickness direction, will be developed. Depending on the elastic properties, the surface friction and the loading conditions there will be slip and no-slip regions in the contact as illustrated in Figure 10. Furthermore, bending deformation of the pin may generate a stress field that varies in the thickness direction of the plate. Thus, the problem is three-dimensional on both a local level, due to the interlaminar stresses, and on a global level due to the bending of the pin. The stress field is affected by the laminate stacking sequence, the size and stiffness of the pin, the hole-pin clearance and friction.

Figure 10: Elastic pin in an orthotropic plate [41].

Accounting for all of these parameters is difficult. In most of the published literature, the interlaminar stresses are neglected and focus is on calculating the 2D stress field in a pin-loaded anisotropic plate. Common approaches are based on either the complex function method or the 2D finite element method. The complex function method was used by de Jong [42] in an early work where he used an assumed stress distribution from the pin, thus transforming the contact problem into a boundary value problem. A more general analysis was conducted by Hyer et. al. [41, 43] in which the complex function method was used for both the pin and the plate, thus

10 accounting for deformation of both parts. The contact problem was solved by means of the collocation method where the slip and no-slip regions were identiﬁed in an iterative process. The 2D ﬁnite element method has been used extensively for this problem [44–51]. In most cases the analyses are straightforward and will not be discussed further. The nonlinear stress-strain relation in shear mode of each lamina was accounted for by Serabian [52] and Shokrieh et. al. [53,54] in 3D analyses, and by several authors [49, 55–59] in 2D analyses. This material behavior is known to be specially important for angle-ply and cross-ply laminates.

Double-lap joint A pin-loaded plate is a laboratory test specimen rather than a real joint. If two splice plates are added, and the pin is replaced by a bolt with washers and a nut, a realistic double-lap joint is created. The main diﬀerence from the pin-loaded plate is that compressive stresses in the thickness direction is generated when torque is applied to the fastener. The plates are thus pressed together which means that some load is transferred by friction between the plates instead of by the bolt-hole contact. The compressive stresses in the thickness direction and the friction between the plates increase the 3D nature of the problem. Some workers focused on calculating the interlaminar stresses while leaving out other parameters such as friction and pin elasticity. A special composite ﬁnite element was developed by Wong and Matthews [60] which allowed for calculations of interlaminar stresses in laminates consisting of a few plies. More complete 3D FE-analyses taking more parameters into account, including interlaminar stresses, friction, pin-elasticity, clamping and clearance have also been conducted [61–64].

Single-lap joint Single-lap joints consist of two member plates and are fundamentally different from pin-loaded plates and double-lap joints. The nonsymmetric geometry of the joint generates a bending moment over the fastener that must be reacted by the contact between the fastener and the plate. This creates a pressure distribution which is nonuniform in the thickness direction and causes an out-of-plane deflection of the joint. This mechanism is referred to as secondary bending and is illustrated for a multi-fastener joint in Figure 11. A severe stress concentration appears at the shear surface of the joint. The magnitude of this stress concentration is affected by several parameters. Clearance will cause the bolt to tilt in the hole when the load is applied. This will initially create a knife-edge like contact. When the load is increased, the contact surface will grow and release the stress concentration. However, the contact surface is likely to be smaller in a joint with clearance as illustrated in Figure 12(a). An other factor is secondary bending. The uneven contact stress causes out-of-plane deflection of the joint. This deflection, on the other hand, improves the alignment of the bolt and the hole surface which will make the contact stress less uneven. Thus, the stress concentration will be reduced if the plates can bend easily. This mechanism is illustrated in Figure 12(b). Furthermore, if different member plates are used there may be a stiffness mismatch, caused by either the geometry of the plates or by their material properties,

11 M M z z x F x

(a) Bending moment over fastener. (b) Moment reacted by uneven contact stress.

Figure 11: Effects of eccentric load path in single-lap joint. which can create different stress concentrations in the two plates. The severe 3D nature of this problem requires that 3D methods are used even if the interlaminar stresses are excluded. This has discouraged workers from analyzing single-lap joints, except in a few cases. Ireman [65] and McCarthy et. al [66,67] conducted 3D finite element analyses of single-lap, single fastener joints. The models included friction, clearance, clamping and both countersunk and protruding head fasteners were analyzed in [65]. Some other authors [68–70] have presented 3D finite element analyses of single-lap joints as part of progressive damage analyses. Papers A-B in this thesis are devoted to investigations of SB. Experimental studies may previously have been prevented by the difficulty in measuring SB on composite plates. Secondary bending is defined according to, ǫ − ǫ SB = b t (3) ǫb + ǫt where the subscripts b and t denotes the bottom and top surfaces of the plate, respectively. Strains are measured with strain gauges on both sides of one plate, close to the hole. Recesses must be machined on the other plate to make room for the gauges and for wiring. This will cause severe damage on the plate which may affect the joint behavior. An alternative technique is developed in Paper-A. Secondary bending is calculated from optical measurements of surface deflections. The method allows for studies of SB without any instrumentation. A numerical study regarding the interaction between secondary bending and bolt-hole contact stress is conducted in Paper-B. Secondary bending is changed in a two-bolt specimen by changing the length of the overlap region. It was found that SB affects both the bearing and net-section stresses, and may have the potential to influence the mode of failure.

12 No clearance Secondary bending

σ σ

Clearance No secondary bending

σ σ

(a) Clearance (b) Secondary bending

Figure 12: Eﬀect of clearance and secondary bending on the contact stress.

Multi-fastener joint The stress field in a multi-fastener joint is more complex. Stresses at a particular hole are affected by the amount of load reacted by the fastener and the amount of load running through the plate cross section which is reacted elsewhere in the joint. These two loads are referred to as bearing load, Pb, and bypass load, Pbp, respectively, and are illustrated in Fig. 13. The load distribution among the fasteners is depending on the fastener pattern, the compliance of each fastener, friction between the plates and in particular by the clearance at each fastener. Clearance at a specific fastener will shift some of its carried load to adjacent fasteners.

bearing

Pbp Pb PA

net − section

Figure 13: Loads acting on a plate in a multi-fastener joint.

The distribution of load among fasteners in a general multi-fastener joint is given by the established force equilibrium when the applied external loads are balanced by the individual bolt loads and friction between the member plates. Simple spring- based methods have been suggested [71, 72] in which fasteners and member plates were considered as springs in parallel and in series, respectively. Assuming force equilibrium and displacement compatibility allowed for solving for the individual fastener loads. These methods are 2D and neglects bolt-hole clearance, fastener clamp-up, friction and out-of-plane deﬂections. Recently, a signiﬁcant improvement to the spring based methods was presented by McCarthy et. al. [73] which enabled initial bolt-hole clearances to be accounted for.

13 Baumann [74] used structural finite elements, i.e. beams and shells, to represent the fasteners and the plates. This approach was further refined by Oakeshott and Matthews [75] in a study on composite double shear joints where composite shell elements were used to model the plates. A more advanced approach was presented by Friberg [76] in which each fastener was represented by a macro element that was placed in square cut-outs in the structure. The macro element was built up from shell elements, beam elements, gap elements and multiple point constraints that represented the plates, the fastener, the fastener-hole contact and the plate-plate contact, respectively. The above methods, with the exception of Fribergs method, are devoted to calculating the load distribution. No information about the stresses in the plates, such as contact stresses or the interlaminar stresses, is obtained. Furthermore, important parameters are omitted from the analyses, including clearance, friction and clamping. A range of 2D analyses, similar to the analyses conducted for pin-joints and single fastener joints described above, have been reported. Methods are based on the 2D finite element method [77–83] or on the method of complex functions [84–86]. Relatively few 3D analyses have been reported on multi-fastener joints. Mc- Carthy and McCarthy [87] developed 3D FE-models based on continuum elements and investigated the influence of bolt-hole clearance on load distribution in single- lap joints with three fasteners arranged in a single column. Good agreement with experiments was found. A sophisticated method capable of efficiently solving the general 3D contact problem for joints with a large number of fasteners was developed by Andersson [88, 89]. The complete global problem needs only to be solved once and subsequent solutions, with modified fastener locations, can be achieved from the global solution at significantly reduced computational cost compared to the global solution itself. This enables the method to be used in parametric and optimization studies which was done for a single-lap joint with 20 fasteners. Friction was not taken into account. The existing analysis techniques for multi-fastener joints can be summarized as follows:

1. Analytical methods and 2D FE-methods are computationally eﬀective and they can account for many of the mechanisms that govern load transfer in multi fastener joints. However, important 3D eﬀects such as fastener clamp- up, friction between member plates and out-of-plane deformations can not be accounted for. Furthermore, general aircraft structures are 3D and thus require methods that are capable of representing such geometries.

2. 3D Finite element methods based on continuum elements provide means to model complex geometries and to take into account all relevant physical mechanisms that aﬀects the load transfer. However, the models are computationally expensive and are in general not suitable for parametric studies or for simulations of large scale geometries.

3. FE-models based on 3D structural elements are computationally eﬀective and capable of modeling complex 3D structures. However, existing techniques con- sistently ignore important factors such as bolt-hole clearance, fastener clamp- up and friction.

14 It appears that a method that is computationally effective and capable of accounting for all important parameters is missing. Part of the work in this thesis is devoted to developing such a method. The approach is to develop a detailed 3D continuum finite element model of a multi-fastener joint that is capable of accounting for all parameters that significantly affect the load distribution. The model is verified by experimentally determined fastener loads. A parametric study is then conducted in order to investigate the importance of different parameters. This work is reported in Paper-C in this thesis. Based on the results in Paper-C, a simplified FE-model based on structural elements is developed. Clearance, friction and fastener clamp-up is modeled by means of special purpose connector elements. The model is verified through comparisons with the detailed model developed in Paper-C. This work is reported in Paper-D in this thesis.

1.4. Strength prediction

Once the stresses are calculated, they can be used in strength analysis. Thus, stresses are evaluated with respect to some failure criterion. A large number of criteria have been developed, usually postulating that the material will fail at a certain level of stress or strain. Using such a criterion to determine the strength of a joint requires that it is used in conjunction with a failure hypothesis. A simple hypothesis would be that the joint has failed when the criterion is fulﬁlled at any point in the laminate. This leads to conservative predictions for joints. More sophisticated hypotheses have been developed, e.g. that joint failure occurs when the criterion is satisﬁed at certain strategic locations or over a certain volume. In this section, some widely used criteria and hypotheses are introduced. It is convenient to distinguish between criteria for the laminate, the lamina, delamination and micro-mechanical failures in the composite.

Laminate based criteria Laminate based methods consider the laminate as a homogenous material and rely on strength data for the laminate. Predictions of net-section failure, based on local stress at the hole edge, has been found to be conservative. One reason for this is that in most stress analyses the material response is assumed to be linear elastic. This results in high stresses in stress concentrations. In the real laminate, the stresses at the concentration is reduced by localized damage, e.g. delamination and matrix cracking. Thus, it is likely that the stresses are overestimated. To overcome this problem, two similar criteria were developed [90]; the point stress criterion (PSC) and the average stress criterion (ASC). The point stress criterion postulates that failure of the joint occurs when the tensile stress at a distance, d0, from the hole edge is equal to the strength of the un-notched laminate. Similarly, the average stress criterion assumes that the laminate fails when the average tensile stress over the distance, a0, equals the un-notched laminate strength. The characteristic distances, d0 and a0, are assumed to be material parameters and must be determined experimentally. However, there is usually a size eﬀect that questions this assumption. It has been found that d0 and a0 may be depending

15 on the size of the hole. A large hole means a larger volume of highly stressed material, compared to the case of a small hole. Larger volume, in turn, implies higher probability that the material contains a large flaw that leads to failure. This phenomenon will affect the real strength of the laminate and thus affect d0 and a0. Pipes et. al. [91] proposed a relation between the hole radius and the characteristic dimensions in order to overcome this problem. More examples of prediction methods related to PSC can be found in [92,93]. Another category of laminate based methods is the fracture mechanics based approach. Waddoups et. al [94] applied linear elastic fracture mechanics (LEFM) to predict strength of composite plates with an open hole. They assumed an energy- intense region of length a to exist at the hole edge. The stress at which fracture occurs is expressed in terms of the characteristic length a, the hole radius and the fracture toughness of the composite. The PSC, ASC and the above LEFM approach all depend on characteristic dimensions that must be determined experimentally. Bäcklund [95] developed a method that only requires basic stiffness and fracture data for the laminate. He suggested that the fictitious crack model (FCM), previously used for failure analysis of concrete, be used to represent the development of damage in the laminate. A fictitious crack was assumed in the high stress region where the stresses are larger than the un-notched strength. The cohesive forces acting on the crack surfaces depends on the crack opening and reduces to zero at some point behind the crack tip. The distance from the hole edge to this point was assumed to represent the length of a real crack. This enabled the modeling of crack growth and the failure load was taken to be the load where the crack propagation turns unstable. Several other authors have presented failure predictions of composite joints based on fracture mechanics [96–98].

Lamina based criteria Lamina based methods are based on strength data for the unidirectional lamina used in the laminate. The advantage compared to laminate based methods is that once the strength data for the lamina has been experimentally determined, failure can be predicted for arbitrary stacking sequences. A disadvantage is that stresses need to be calculated in each lamina, compared to the laminate based methods which only requires stresses for the homogenized plate. Typically, a stress (or strain) envelope in 2D or 3D is assumed to define failure of the lamina. Lamina criteria can be used either to predict the first failure of any lamina, so called first-ply failure, or to calculate the failure of the entire laminate, so called last-ply failure. In the latter case, stresses are calculated in all laminae and the criterion is applied for each lamina. Laminae that fails are excluded from the analysis and a new stress state is calculated. This process is repeated until all laminae have failed. The Tsai-Wu criterion [99] assumes the existence of a failure surface, f(σk), in the three dimensional stress space that can be expressed as

f(σk)= Fiσi + Fijσiσj =1 (4) where σi are components of the stress tensor, Fi and Fij are strength tensors, i, j, k = 1 . . . 6 and the Einstein summation convention is used. It is assumed that

16 the strength tensors, Fi and Fij, are symmetric in accordance with the stress tensor, and the elastic compliance tensor, which implies that the number of independent entries in the tensors are 6 and 21, respectively, for a general anisotropic material. Further symmetry in the material reduces the number of entries in the strength tensors in analogy with the reduction of entries in the compliance matrix. All non-zero entries must be determined through a series of uni-axial and bi-axial tests. The linear terms, Fi, implies that the different strengths in tension and compression of a composite is accounted for. The quadratic terms, Fij, defines an elliptic failure surface in the stress space where the sensitivity to multi-axial stresses is captured when i =6 j. Similar criteria were developed by several authors [2]. Most of them are simpler and requires fewer material parameters. The criterion developed by Hoffman [100] has been used in many studies to predict strength of joints. A simpler, in-plane quadratic criterion, suggested by Yamada and Sun [101] has been used extensively for bolted joints. It involves only two material parameters and is defined according to 2 2 σ1 τ12 + =1 (5) X S12 where X and S12 are the longitudinal ply strength and the ply shear strength, respectively. The absence of linear terms implies that the material is assumed to have equal strength in tension and compression. Maximum stress, Maximum strain and the Distortional energy criteria were used by Waszczak and Cruse [102] in an early study. Each criterion was used with the simple hypothesis that a pin-joint would fail when the criterion was satisfied in the vicinity of the hole. This proved to be conservative in most cases. Soni [103] used the Hoffman criterion [100] in conjunction with a more advanced failure hypothesis. The 2D stress components from the stress analysis were used in the criterion to calculate the maximum average bearing stress that could be carried by each lamina. The lamina capable of sustaining the highest stress was considered to be the strongest lamina. Ultimate failure of the joint was assumed to occur when the strongest lamina failed. Laminae failing prior to ultimate failure were not removed from the analysis, i.e. no stiffness reduction of the laminate due to localized damage was accounted for. This method still proved to be conservative. Improved accuracy was achieved in a method suggested by Chang, Scott and Springer [11]. The Yamada-Sun criterion [101] was used with a failure hypothesis that can be considered as an extension of the laminate based point stress criterion. A characteristic curve was constructed from R0t and R0c which denotes characteristic length in the net-section and bearing plane, respectively. Thus R0t is identical to d0 that was used in [90]. Such a curve is illustrated in Figure 14. The failure criterion is evaluated along the characteristic curve and the joint was considered to fail when the criterion was satisfied for any lamina, i.e. it is a first-ply failure prediction. The location on the curve where failure was predicted determines the failure mode to be respectively, bearing, shear-out and net-section according to −15◦ 6 θ 6 15◦ 30◦ 6 θ 6 60◦ ◦ ◦ 75 6 θ 6 90 .

17 Figure 14: Characteristic curve [11].

This method provided accurate predictions despite the fact that a lamina based failure criterion was used with laminate based characteristic dimensions that are obtained in uni-axial testing. The combination of the proposed characteristic curve and the Yamada-Sun criterion has been used in subsequent analyses [104–109]. The Hoﬀman criterion was used by DiNicola and Fantle [110] in conjunction with a characteristic distance, dfe, similar to R0c on the characteristic curve discussed above, to predict bearing failure in a pin-joint. It was proposed that dfe be dependent on hole size and stacking sequence which resulted in accurate predictions.

Delamination criteria A criterion for delamination onset in composites was formulated by Ye [111]. He suggested that delamination is controlled by the interlaminar stress state, regardless of the nature of the external loading, and proposed the following criterion,

2 2 2 σ33 τ13 τ23 + + =1 σ33 > 0 (6) Z S13 S23 2 2 τ13 τ23 + =1 σ33 < 0 (7) S13 S23 where Z is the transverse tensile strength and Sij are the corresponding shear strengths. The stresses in Eq. 6 are deﬁned as average stresses over a distance c from the free edge. Ye suggested c to be equal to the thickness of two plies. This criterion has been used by several authors [62–64] to predict delamination in joints.

Micro-mechanics based criteria Failure in composite materials is characterized by accumulated damage in several micro-mechanical failure modes, such as fiber and matrix fracture, fiber pull-out, matrix micro-cracks and fiber-matrix debonding. Laminate and lamina based failure theories are extensions of theories developed for isotropic metals. Therefore

18 they are not closely related to the actual physical mechanisms occurring in the material. They all have in common that they predict the complete failure of the entire laminate or lamina. No distinction is made between the different micro-mechanical failure modes, and the laminate or lamina is typically assumed to have zero stiffness after failure. An approach closer related to the actual physics involved, is to predict the micro-mechanical mode of failure and change the macroscopic properties of the lamina based on this prediction. Such progressive damage analyses have been conducted by several authors [12,14,53,55–59,69,70,112–115]. A widely used set of criteria for tensile and compressive failure of both the matrix and the fibers was formulated by Hashin [116]. The steps required in a progressive damage analysis are illustrated in Figure 15.

Start

Stress analysis

Failure analysis

No Check for Increase ply Load failures

Yes

Degradation of material properties on failed plies

No Check for ﬁnal failure

Yes

Stop

Figure 15: Flow of a progressive damage analysis. Redrawn from [70].

19 Progressive damage analyses are important in order to understand the failure process and to calculate the maximum load capacity of a joint under an overload situation. However, such analyses are complicated and computationally expensive. From a design point of view it is also important to maximize the load capacity of the joint with respect to the introduction of damages. Thus it becomes important to know what is the initial mode of failure and how can it be predicted. This is particularly so for the macroscopic bearing mode. It is well known that damages may be introduced well before the ultimate failure of the joint. Bearing failure [10,117,118] is developed by the compressive forces acting between the fastener and the hole surface. It is a crushing mode which involves several micro- mechanical failure modes, usually starting with matrix cracking followed by buckling of destabilized fibers. This buckling starts in the 0◦-plies, in the bearing plane, due to their great stiffness in the loading direction. As loading continues to increase, delamination starts to occur and kink bands of buckled fibers are formed, as shown in Figure 16.

(a) Joint loaded to 23 kN. (b) Close-up of kink band in a 0◦-ply.

Figure 16: Formation of kink bands in the bearing plane.

These kink bands eventually lead to shear cracks progressing through all plies, from the center of the laminate towards the surface, at a 45◦ angle. If the cracks reach the surface under the washer, it means that the damaged material is stabilized by the clamping pressure from the washers. This may enable the joint to carry more load. Eventually, a second set of shear cracks are formed at the center of the laminate outside the first set. The new cracks will reach the laminate surface outside the washer and cause ultimate failure of the joint. This stage-by-stage formation of shear cracks is typical for joints [10] and is illustrated for a double-lap joint in Figure 17. Several workers have suggested criteria for fiber micro-buckling. A 2D-model of fibers embedded in a matrix was developed by Rosen [119], where the fibers are assumed to buckle in the elastic foundation at the critical compressive stress, σc, according to

Gm σc = = G (8) 1 − vf where vf the ﬁber volume fraction, Gm is the shear modulus of the matrix and G is the shear modulus of the lamina. Rosen found that the model overestimates the

20 (a) Joint loaded to 26 kN. (b) Joint loaded to 29 kN.

Figure 17: Bearing damages in double lap specimens subjected to diﬀerent levels of tension loading.

φ0 + φ σ σ

Figure 18: Deﬁnitions of angles and stresses.

compressive strength for many materials. This can be attributed to the assumptions that the ﬁbers are perfectly aligned in the loading direction and that the matrix material is linear elastic. In reality, a large number of ﬁbers are misaligned and strains in the matrix material exceeds the yield limit which implies that the plastic response is important. Argon [120] developed an expression for compressive strength according to,

τy σc = (9) φ0 where τy is the plastic shear strength of the composite and φ0 is the original angle of misalignment of the fibers. It was recognized that when the compressive stress exceeded σc, a local instability in the form of a shear collapse band, or kink band, developed. This expression considers plastic micro-buckling of the fibers as it involves the plastic shear strength instead of the elastic shear modulus of the matrix. Budiansky [121] generalized Eq. 9 in order to take into account the angle, β, at which the kink band is inclined with the fiber direction, see Fig. 18, according to

τy/G 2 σc = (G + Ezz tan β) (10) φ0 + τy/G where Ezz is the transverse modulus of elasticity. Further generalization was provided by Slaughter et. al. [122] such that a multi-

21 axial stress state was accounted for according to

ατy − τxz − σzz tan β σc = (11) φ0 + φ

2 σzz α = 1+ y tan2 β (12) τ s y where σzzy is the transverse yield strength and φ is the additional rotation of fibers within the kink band. This micro-buckling criterion is 2D in the sense that the width of the material is not accounted for. To the best of the authors knowledge, it has not been used to predict strength of bolted joints. In Paper-E in this thesis, the criterion is used in conjunction with a simple FE-model and a failure hypothesis to predict strength of a multi-fastener single-lap joint. The FE-model is based on structural elements and calculates the load distribution while accounting for clearances, clamping, friction and secondary bending. Bolt-hole contact stresses are obtained from simple force and moment equilibrium equations, and assumptions regarding the stress distribution in the thickness and circumferential directions. The failure hypothesis is that the joint has failed when the criterion is fulfilled in the most highly stressed lamina, in the joint bearing plane, of the most heavily loaded fastener. This is assumed to correspond to the first irregularity in experimental load-displacement curves.

1.5. Future work

The use of composite materials in aircraft is steadily growing. Still, the full weight- saving potential of such materials may not be reached due to suboptimal design of the joints. There is a constant need for improved design methods, both on a global level and on a local level. On the global level, joints are limiting factors of the entire structure. The behavior of the joints should be accounted for as early as possible in the design process. Finding optimum bolt patterns is vital to the final performance of the structure. This requires effective, accurate predictions of load distribution and strength in large scale structures with many bolts. The ability to use such predictions in optimization routines may improve the overall design. Statistical analyses of “missing bolt” or “worst case clearance” scenarios may increase the ability to design fail safe structures. A starting point for such a method is presented in Papers D-E in this thesis. For industrial use, the method should be extended to use shell elements to represent the plates, to enable modeling of complicated structures. Further extensions could be to use a more sophisticated procedure to calculate the stresses, based on the forces and moments calculated in the FE-model, and to apply several failure criteria to predict other modes of failure. On the local level, strength prediction of critical fasteners remains an issue. De- spite large efforts in this field, accurate and fully reliable methods are not available. This results in conservative designs with a weight penalty. While semi-empirical methods and laminate- and lamina based methods continues to play important roles

22 in todays design procedure, methods based on micro-mechanical mechanisms seem more promising in terms of improved accuracy in predictions. This should be re- ﬂected in future research activities within this ﬁeld.

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