COMBINED IN-PLANE AND THROUGH-THE-THICKNESS ANALYSIS FOR FAILURE PREDICTION OF BOLTED COMPOSITE JOINTS

COMBINED IN-PLANE AND THROUGH-THE-THICKNESS ANALYSIS FOR FAILURE PREDICTION OF BOLTED COMPOSITE JOINTS V. Kradinov, * E. Madenci The University of Arizona, Tucson, Arizona, 8575 D. R. Ambur NASA Langley Research Center, Hampton, Virginia, 2368 Abstract Alough two-dimensional meods provide accurate predictions of contact stresses and bolt load distribution in bolted composite joints wi multiple bolts, ey fail to capture e effect of icness on e streng prediction. Typically, e plies close to e interface of laminates are expected to be e most highly loaded, due to bolt deformation, and ey are usually e first to fail. This study presents an analysis meod to account for e variation of stresses in e icness direction by augmenting a two-dimensional analysis wi a onedimensional rough e icness analysis. The twodimensional in-plane solution meod based on e combined complex potential and variational formulation satisfies e equilibrium equations exactly, and satisfies e boundary conditions and constraints by minimizing e total potential. Under general loading conditions, is meod addresses multiple bolt configurations wiout requiring symmetry conditions while accounting for e contact phenomenon and e interaction among e bolts explicitly. The rough-eicness analysis is based on e model utilizing a beam on an elastic foundation. The bolt, represented as a short beam while accounting for bending and shear deformations, rests on springs, where e spring coefficients represent e resistance of e composite laminate to bolt deformation. The combined in-plane and rough-e-icness analysis produces e bolt/ hole displacement in e icness direction, as well as e stress state in each ply. The initial ply failure predicted by applying e average stress criterion is followed by a simple progressive failure. Application of e model is demonstrated by considering single- and double-lap joints of metal plates bolted to composite laminates. *Technical Expert, Department of Aerospace and Mechanical Engineering. Professor, Department of Aerospace and Mechanical Engineering. Member AIAA. Head, Mechanics and Durability Branch. Associate Fellow AIAA. Copyright 24 by e American Institute of Aeronautics and Astronautics. All rights reserved. Introduction Bolts provide e primary means of connecting composite parts in e construction of aircraft and aerospace vehicles. The main disadvantage of bolted joints is e formation of high stress concentration zones at e locations of bolt holes, which might lead to a premature failure of e joint due to net-section, shear-out, or bearing failures, or eir combinations. The stress state in a bolted joint is dependent on e loading conditions, dimensions, laminate stacing sequence, bolt clamp-up forces, bolt location, bolt flexibility, bolt size, and bolole clearance (or interference). A substantial number of experimental, analytical, and numerical investigations have been conducted on e stress analysis of bolted laminates. The study by Kradinov et al. provides an extensive and detailed discussion of earlier investigations. In order to eliminate e shortcomings of e previous analyses, Kradinov et al. introduced a twodimensional numerical/analytical meod to determine e bolt load distribution in bolted single- and doublelap composite joints utilizing e complex potentialvariational formulation. This meod addresses multiple bolt configurations wiout requiring symmetry conditions while accounting for e contact phenomenon and e interaction among e bolts explicitly under bearing and by-pass loading. The contact stresses and contact regions are determined rough an iterative procedure as part of e solution meod. Alough is two-dimensional approach provides an accurate prediction of e contact stresses and bolt load distribution, it fails to capture e effect of icness on e failure prediction. In addition to e head and nut shapes and e applied bolt torque, e stacing sequence considerably influences e stress state in each ply of e laminate. Thus, an adequate representation of e ply load variation rough e icness is critical for e failure prediction of composite laminates at e bolt-hole boundary. This study presents an analysis meod to account for e variation of stresses in e icness direction by augmenting e two-dimensional analysis by Kradinov et al. wi a model of a bolt on an elastic foundation, as suggested by Ramumar et al. 2 The bolt, represented as a beam while accounting for bending and shear deformations, rests on springs, where e spring coefficients

represent e resistance of e composite laminate to bolt deformations. The values of e spring coefficients depend on e fiber orientation of e laminate plies; for isotropic plates, e spring coefficients are defined by a constant value. The present analysis produces e bolt/ hole displacement in e icness direction and e stress state in each ply. Failure load and associated failure modes of net-section, bearing, and shear-out for composite bolted joints are predicted based on e average stress criterion of Whitney and Nuismer 3 for first ply failure, followed by a simple progressive failure criterion as suggested by Ramumar et al. 2 The applicability of is meod is demonstrated by considering single- and double-lap joints of laminates wi a varying number of bolts. In addition to e determination of e contact stresses and e bolt load distributions, e failure load is investigated by applying a progressive failure procedure based on e average stress failure criterion. Problem Statement The geometry of bolted single- and double-lap joints of composite laminates is described in Fig.. Each joint can be subjected to a combination of bearing, by-pass, and shear loads. Each laminate of e single- and double-lap joints, joined wi L number of bolts, can be subjected to tractions and displacement constraints along its external boundary. The icness of e laminates is denoted by h. As illustrated in Fig. 2, e hole radius in e laminate associated wi e l bolt, a, l (which is slightly larger an e bolt radius, R l ), leads to a clearance of δ, l. The ranges of e subscripts are specified by,..., K and l,..., L, wi K and L being e total number of laminates and bolts, respectively. The bolt radius remains e same in each laminate; however, e radii of e holes associated wi e same bolt are not necessarily e same. The extent of e contact region is dependent on e bolt displacement deformation of e hole boundary, and e clearance. The presence of friction between e bolts and e laminates is disregarded. Each laminate wi a symmetric lay-up of N plies can have distinct anisotropic material properties. Each bolt can also have a distinct stiffness, and e explicit expressions for bolt stiffness for a single- and double-lap joint, as well as e general lap configurations, are derived in Kradinov et al. The problem posed concerns e determination of e extent of e contact zones, e contact stresses and e bolt load distribution under general loading conditions, e bolt/hole deformation, and e stress state in each ply, and us e joint streng. Fig. Geometric description of single- and doublelap bolted joints. Fig. 2 Position of a bolt before and after e load is exerted. Solution Meod In-Plane Analysis for Contact Stresses and Bolt Load Distribution The coupled complex potential and variational formulation introduced by Kradinov et al.,4 is employed to determine e two-dimensional stress and strain fields required for e computation of e contact stresses and contact regions, as well as e bolt load distribution. This in-plane analysis is capable of accounting for finite laminate planform dimensions, uniform and variable laminate icness, laminate lay-up, interaction among bolts, bolt torque, bolt flexibility, bolt size, bolt-hole clearance and interference, insert dimensions, and insert material properties. Unlie e finite 2

element meod, it alleviates e extensive and expensive computations arising from e non-linear nature of e contact phenomenon. Also, e meod is more suitable for parametric study and design optimization. Alough is two-dimensional analysis provides accurate in-plane stresses in each laminate and bolt load distribution, it assumes no variation of stresses rough e laminate icness. This assumption might lead to erroneous results in e streng prediction of bolted joints because of e pronounced influence of roughe-icness stress variation at e bolt location as discussed by Ramumar et al. 2 Through-Thicness Analysis for Bolt/Hole Deformation In conjunction wi a two-dimensional in-plane bolted joint analysis, Ramumar et al. 2 suggested a model utilizing a beam on an elastic foundation in order to include e variation of stresses in e icness direction of e bolted joint. The bolt rests on springs, where e spring constants represent e resistance of e laminate to bolt deformation. The spring constants correspond to e modulus of each ply rough e icness of e hole boundary. Their values depend on e ply orientation of e laminate. For isotropic plates, e spring constants have a uniform value. As e bolt bends, e plies are loaded differently near e hole boundary based on eir orientation and location. As shown in Fig. 3, e plies close to e interface of adjacent laminates exhibit significant deformation. As shown in Fig. 4, e beam representing e bolt rests on an elastic foundation whose modulus is represented by e stiffness of each ply, i,, in e laminate. The superscript l and subscripts and i denote e specific bolt, e laminate, and e ply numbers, respectively. Also, e bolt is subjected to constraints at e head and nut locations rough rotational stiffness constants, l h and n, in order to include e effect of head and nut shapes and bolt torque. Free-body diagrams of e laminates at e l bolt in a single- and double-lap joint and e end conditions and slope continuity conditions in e presence of bo bending and shear deformations are shown in Fig. 5. In accordance wi e typical bolt deformation illustrated in Fig. 3, e force exerted by e l bolt on e laminate, P l (obtained from e two-dimensional inplane analysis), is enforced as a shear force, V, at e interface of e adjacent laminates. At e interface, e continuity of e bending slopes, ψ ψ +, is also enforced while permitting e laminates to displace. At e head and nut locations of e bolt, e shear force values are set to zero, and e rotations (slopes) are dictated by rotational stiffness constants, h and l n, depending on e bolt type, presence of washers, and e applied bolt torque. Finite Element Analysis. The bolt/hole displacements rough e laminate icness are obtained by discretizing e bolt wi beam elements at account for bending and shear deformations. The bolt discretization is based on e discrete nature of e ply stacing sequence. Along its icness, each ply is discretized wi two beam elements. For bo single- and doublelap joints, e number of elements and e number of nodes in relation to e number of plies in e laminate are described in Fig. 6. In e discretization process, a node located in e middle of each ply is attached to a spring element representing e ply stiffness (Fig. 6). The derivation of e stiffness matrix composed of a two-noded beam element (Timosheno s zero-order shear deformable beam eory) and a linear spring element is presented in e Appendix. Associated wi e laminate and e l bolt, ( each node is assigned a deflection, l), j ( z j), and ( l a rotation, φ, j φ ) ( z j), wi e subscript j representing e node number. In e finite element formulation, e rotations of e internal nodes are statically condensed in terms of e nodal displacements and e end node rotations. The positive directions of e deflections and rotations are shown in Fig. 7. The details of e condensation procedure are also explained in e Appendix. (a) (b) Fig. 3 Typical bolt deformation in a (a) single- and (b) double-lap joint. 3

(a) (a) ( b) Fig. 4 Bolt on an elastic foundation model in a (a) single- and (b) double-lap joint. (b) Fig. 5 Free-body diagrams of laminates for a (a) single- and (b) double-lap joint. 4

(a) (b) Fig. 6 The finite element model of a bolt in a (a) single-and (b) double-lap joint. Fig. 7 Bolt discretization in e plate after static condensation. 5

Spring Stiffness Coefficients. As suggested by Ramumar et al. 2, e translational spring stiffness coefficients, i,, representing e i ply of e laminate near e l bolt are approximated by p i, i, γ, i, N () l where p l is e load exerted by e l i, bolt on e i ply of e laminate and γ l represents e maximum hole enlargement of e l hole in e lami- nate. The number of plies in e laminate is denoted by N. As part of e two-dimensional in-plane bolted joint analysis, e load exerted by e bolt on e i ply of e laminate near e l hole, p l, is computed as i, 2 2 i, ( i, ) x ( i, ) y p p + p (2) where p l ( ix, ) and p ( i, ) y represent its components in e x- and y-directions. These components are computed by integrating e radial stresses in each ply as and 2π ( i, ) ( i, ) x, rr, p a l σ ( r a l, θ) cosθ dθ (3a) 2π ( i, ) ( i, ) y, rr, p a l σ ( r a l, θ) sinθ dθ (3b) in which a, l is e radius of e l hole in e ( i, ) laminate, and σ rr ( r a, l, θ ) represents e radial stress distribution in e i ply of e laminate near e l hole. Under plane-stress assumptions, e stress and strain components are related by σ Q, ε (4) ( i, ) i in which Q i, represents e reduced stiffness matrix for e i ply of e laminate. The Cartesian stress components in e i ply of e laminate and e Cartesian strain components, uniform rough e icness of e laminate, are included in e vectors of σ and ε. This stress state in each ply ( i, ) is employed in e prediction of e initial ply failure load, IN F l i,, and e corresponding failure mode. As shown in Fig. 2, e maximum hole enlargement, γ l, is defined as e absolute value of e difference between e radial displacements, u ( r a, l, θ θ) and u ( r a,, θ θ ) l 2,of points and 2 on e hole boundary in e direction of e bolt load γ u ( r a, θ θ ) u ( r a, θ θ ) (5), l 2, l where e radial displacements are obtained from e in-plane bolted joint analysis. The maximum holeenlargement, γ l, can be different for each laminate of e bolted joint, but is uniform rough e icness. Also, it is specific to each bolt-hole in e laminate because it is dependent on e deformation response and e bolt load distribution. The head and nut ( l ) rotational stiffness coefficients, h and l n, respectively, have values close to zero for free-end conditions and to infinity for protruding head bolts under high torque. The stiffness matrix becomes singular if ese coefficients approach zero. The analysis results include displacements and rotations, l, j and φ l, j, at e j node, as well as e spring forces, f l i,, at e i ply of e laminate near e l hole. The effect of rough-e-icness variation is invoed in e in-plane stress analysis by considering e spring forces, f l i,, as e corrected ply loads. Progressive Failure Prediction There are ree major failure modes in bolted composite lap joints: net-section, shear-out, and bearing (Fig. 8). The net-section failure is associated wi fiber and matrix tension failure and shear-out and bearing failures are associated wi fiber and matrix shear and compression failures, respectively. Failure in bolted laminates can be predicted by evaluating eier e specific stress components or eir interaction at characteristic distances from e hole boundary. Alough any one of ese criteria is applicable to e prediction of e failure of a laminate or a ply, values of e characteristic distances and e unnotched streng parameters of e material are scarce. Fig. 8 Primary failure modes in bolted composite joints. 6

The point and average stress criteria introduced by Whitney and Nuismer 3 disregarded e interaction among e stress components. However, ey are widely used in engineering practice for predicting e failure stress and failure modes because of eir well-established values of e characteristic distances. 2,5,-6 Bo of ese criteria predict net-section, shear-out, and bearing failures when e stress components at specific locations reach eir corresponding unnotched streng ns levels. The characteristic distances of a o for netsection, a o br so for bearing, and a o for shear-out failures, as well as e shear-out and net-section planes (denoted by e n and s lines), are shown in Fig. 9. According to e point stress criterion, e net-section failure occurs ns when e normal stress, σ ss, at a distance a o from e bolt-hole boundary along e net-section plane reaches e unnotched tensile streng of a ply, X t. If σ ss at a br distance a o from e bolt hole boundary reaches e unnotched compressive streng of a ply, X c, bearing failure occurs. Shear-out failure occurs when e shear stress, σ ns, at a distance a so o from e bolt-hole boundary along e shear-out planes reaches e unnotched shear streng of a ply, X s. The average stress failure criterion is based on e average values of e corresponding stress components over e characteristic distances of ns ) br so (, a, (, a ), and (, a ). Under e specified external loading, e ratios (, l) Cji, ( j ns, br, so) of e unnotched streng parameters to e average stresses associated wi e netsection ( j ns), bearing ( j br), and shear-out ( j so) failure modes at e i ply of e laminate near e l hole are defined in e form C X ( i, ) (, l) t ns, i ( i, ) σ ss for net-section failure mode (6a) C C X ( i, ) (, l) c br, i ( i, ) σ ss X ( i, ) (, l) s so, i ( i, ) σ ns for bearing failure mode for shear-out failure mode (6b) (6c) in which ( i, ) (, ) σ ss and σ i ns are averaged normal and ns shear stresses over e characteristic distances (, a ), br so (, a ), and (, a ). IN The initial ply failure load, F i,, and its associated failure mode are established by IN, min (, ) l (, l) i ji i, F C p, ( j ns, br, so) (7) After e initial failure, a ply is assumed to continue sustaining e applied load according to a bilinear behavior, shown in Fig.. The value of e ultimate UL ply failure load, F l, is defined by i, F UL i, HF l IN i, (8) where e factor H varies as.2,.5, and.2 for net-section, bearing, and shear-out failure modes, respectively, as suggested by Ramumar et al. 2 Due to e bilinear ply load behavior, e applied joint load is increased incrementally while predicting ply failure subsequent to e initial ply failure. At each load increment, e corrected ply loads, f ( l ) i,, are compared to e initial and ultimate ply failure loads of IN F l UL i, and F l i,, which are predicted according to e average stress criterion of two-dimensional analysis. Fig. 9 Characteristic distances for point and average stress failure criteria. Fig. Bilinear ply behavior. 7

For an undamaged ply, if e corrected ply load of f l i, exceeds e corresponding initial ply failure load IN of F i,, e ply experiences initial failure. Accordingly, as suggested by Ramumar et al., 2 e initial ply stiffness of is reduced to ˆ i, i,. The reduced ply stiffness, ˆ, is defined by i, ˆ α i, i, (9) in which e parameter α is assumed to be.. For a damaged ply, if e corrected ply load of f l i, exceeds UL e corresponding ultimate failure load of F i,, e ply experiences total failure. Consequently, e ply stiffness is reduced to zero. Based on e bilinear behavior of e ply load shown in Fig., e ply load at e i ply of e laminate near e l bolt can be expressed in terms of e ply displacement as Fig. One-bolt single-lap joint geometry and loading. f l i, i, j (a) for an undamaged ply, as for a damaged ply, and as ˆ IN ˆ, l f + (b) ( l ) ( l ) ( l ) ( l ) i, i, i,, j i, j f i, (c) for a totally damaged ply, where j denotes e node associated wi e translational spring element representing e i ply. When a ply fails, e adjacent plies share e load released by e failed ply. Thus, e failure propagates from ply to ply until e total failure of e laminate. The ultimate joint failure load is defined as e joint load at results in e ultimate failure of half of e plies at a particular bolt location. The minimum of e failure loads predicted for each bolt establishes e streng of e joint. This type of progressive failure analysis can be employed in conjunction wi any one of e available failure criteria. Numerical Results The capability of is combined in-plane and rough-e-icness bolted joint analysis is demonstrated by considering single- and double-lap bolted joints joining metal to composite laminates wi one, ree, and four bolts as shown in Figs. -3. The material properties, stacing sequence and icness of e plates are e same as ose considered by Fig. 2 Three-bolt double-lap joint geometry and loading. Fig. 3 Four-bolt single-lap joint geometry and loading. 8

Ramumar et al. 2 The metal plates are made of aluminum wi Young s modulus Ea. Msi and Poisson's ratio ν a.3. The bolts are of steel wi a Young's modulus E s 3. Msi and Poisson's ratio ν s.3. Alough not a limitation of e analysis meod, in ese configurations, e bolt and hole diameters are equal, leading to zero clearance. The aluminum plates have a icness of.3 in. The icness of e laminate is.2 in wi stacing sequence of [(45 / / 45 / ) 2 / 9 ] s. The material properties for each ply are specified as E L 8.5 Msi, E T.9 Msi, G LT.85 Msi, and ν LT.3. The high value of torque applied on e protruding bolt-head is specified by e head and nut rotational stiffness coefficients of 2 2 h lbs-in and n lbs-in. The failure prediction is performed by employing e average stress criterion along wi a bilinear stiffness reduction after e initial failure of each ply. The characteristic leng parameters for e average stress failure criterion are taen as a ns so. in, a.8 in, and br a.25 in. The unnotched streng parameters of e ply for each orientation in e stacing sequence are given in Table. Fig. 4 Stress variation around e hole boundary in an aluminum plate. Table Unnotched streng values in X-direction. Ply X t, Netsection Bearing X X c, s, Shearout orientation (degree) tensile (si) (si) (si) 23. 32. 7.3 45 4. 56. 95. -45 4. 56. 95. 9 9.5 38.9 7.3 As part of e finite element modeling, e section of e bolt in contact wi e composite laminate is discretized wi 4 nodes in order to represent 2 plies of e laminate lay-up. Because e aluminum plate is icer an e laminate, it is discretized wi 8 nodes leading to 4 layers of aluminum. One-Bolt Single-Lap Metal to Composite Joint The geometrical parameters shown in Fig. are defined by W.875 in, d.325 in, L 3.6 in, L2 4.4375 in, and s.9375 in. The initial applied load of P 875 lbs is uniformly distributed along one edge of e aluminum plate while e oer end of e laminate is constrained. The variations of radial and tangential stresses around e hole boundary in aluminum and composite plates are shown in Figs. 4 and 5, respectively. These figures demonstrate e capability of e two-dimensional analysis to capture e stress concentrations and provide e contact region around Fig. 5 Stress variation around e hole boundary in a composite laminate. e bolt hole. The segment of e radial stresses wi negative values establishes e contact region between e bolt and hole boundary. As expected, ere are zero shear stresses on e hole boundary because of e absence of friction. Based on e in-plane stress analysis, e maximum () -4 hole enlargements are computed as γ 6.88 in () -4 and γ 2 2.89 in for e aluminum and laminate, respectively. Invoing ese values in Eq. (), e stiffness of e spring representing e aluminum layer () has a value of, i 69, 597 lb/in wi i, 4. The spring stiffness value for each ply of e laminate is 9

() calculated as () 2, m 34, 286 lb/in, 2, n 52,586 lb/in, () () 2, p 34, 654 lb/in, and 2, q 8, 446 lb/in, where e subscripts m, n, p, and q represent 45,, -45, and 9 plies, respectively. () The variation of e nodal displacements,, j wi () ( j,8) and 2, j wi ( j,4), illustrates e bolt/hole deformations in Fig. 6. As observed in is figure, e deformations in e composite laminate are larger an ose in e metal plate as dictated by e material properties and laminate icness. As expected, e specified large values for head and nut rotational stiffness coefficients, h and n, result in zero slopes at e ends of e bolt. The maximum bolt/hole deformations occur at e interface of e two plates, indicating e location of e major load transfer, as reflected in Fig. 7, which depicts e variation of e load distribution rough e icness of e joint. As expected, e load distribution rough e icness of aluminum plate varies continuously. However, e ply loads corresponding to e composite laminate change abruptly, depending on e fiber orientation. This behavior is dictated by e material property discontinuity in e icness direction resulting in a different stress state in each ply. As presented in Table 2, e initial ply failure is predicted at a load level of P 3,656 lbs, wi a netsection failure mode in ply number wi a 9 fiber orientation. As e applied joint load is increased incrementally, e plies wi a 9 fiber orientation continue failing in e net-section failure mode. Their failure is followed by a mixture of ±45 and plies in e netsection and bearing failure modes, respectively. The load increments resulting in no failure have been omitted in Table 2. At load increment 56, ply number wi a 45 fiber orientation ultimately fails at a load level of P 5,336 lbs. This ply failure is followed by eleven different ply failures at e same load level. Therefore, e ultimate joint failure is reached at load increment 67 at a load level of 5,336 lbs. This prediction is in acceptable agreement wi e experimental measurement of 4,9 lbs reported by Ramumar et al. 2 Three-Bolt Double-Lap Metal to Composite Joint The geometrical parameters shown in Fig. 2 are defined by W 2.5 in, L 3.6 in, L 2 4.525 in, s.25 in, s2. in, s 3.9 in, s 4.8 in, h.25 in, h 2.5 in, h 3.4 in, and d.325 in. The initial joint load of P 25 lbs is applied to e aluminum plate while e ends of e composite laminates are constrained. The maximum hole enlargement values associated wi each bolt hole are computed from e two-dimensional analysis and are presented in Table 3, and e spring stiffness values for each ply are in Table 4. The rough-e-icness variation of e ply loads near bolt number 3 is shown in Fig. 8. The corresponding bolt/hole deformations are depicted in Fig. 9. As observed in ese figures, e most pronounced deformation occurs in plies located along e plate interfaces. Bo deformations and ply load distributions are identical for composite laminates due to e presence of symmetry in e material and geometry. Bolts 2 and 3 exert higher loads on e composite an bolt. The sequence ply failure loads and modes associated wi each bolt are different because of e different strain states in e laminate near each bolt hole. As presented in Table 5, e initial ply failure near bolt occurs at a load level of 22,42 lbs, in ply number 9 wi a 45 fiber orientation, in e shear-out failure mode. Part of e laminate near bolt becomes unstable at load increment 23, corresponding to a load of 24,745 lbs, in ply wi a 9 fiber orientation, in e net-section ultimate failure. At is load level, seventeen more failures occur in e composite laminate before e laminate is assumed to ultimately fail at load increment 39. As presented in Tables 6 and 7, e initial ply failures near bolts 2 and 3 occur at 3,24 lbs and 2,769 lbs, respectively, in ply wi a 9 fiber orientation, in e net-section failure mode. The progress of failure near bolt 2 is presented in Table 6. Starting at load increment 46 and until 64, failure occurs for nineteen increments in different plies at a load of 8,225 lbs, and e joint can still carry more load. Finally, ultimate failure of e joint occurs at load increment 69, corresponding to a load level of 9,55 lbs, in ply 2 wi a fiber orientation, in shear-out ultimate failure. A similar failure behavior is observed near bolt 3, as presented in Table 7. At load increment 24, corresponding to a load of 3,965 lbs, failure occurs in ply wi a 45 fiber orientation, in e net-section failure mode, followed by fifteen failures in different plies at e same load level until ultimate joint failure. Thus, e ultimate joint failure load is computed as 3,965 lbs near bolt 3. As shown in Table 7, e sequence of ply failure indicates at 9 and ±45 plies fail wi e net-section failure mode while plies fail wi e shear-out failure mode. Four-Bolt Single-Lap Metal to Composite Joint The geometrical parameters for e four-bolt doublelap joint shown in Fig. 3 are defined by W 3.25 in, s.25 in, e.9375 in, l 2.75 in, and D.325 in.

Fig. 6 Variation of bolt/plate displacement rough e joint icness. Fig. 7 Variation of ply loads rough e joint icness.

Load increment Table 2 Progressive ply failure in one-bolt single-lap joint. Ply Applied joint Ply orientation, load, (lb) number (degree) Failure mode 3656 9 net-section 3 3693 9 net-section 9 4287 9 net-section ultimate 22 4373 9 net-section ultimate 34 4879 45 net-section 37 4977 3-45 net-section 39 527 5 45 net-section 4 577 2 bearing 43 528 4 bearing 44 528 7-45 net-section 46 579 6 bearing 48 523 8 bearing 5 5283 9 bearing 5 5283 4-45 net-section 52 5283 6 45 net-section 53 5283 8-45 net-section 54 5283 2 45 net-section 56 5336 45 net-section ultimate 57 5336 3-45 net-section ultimate 58 5336 5 45 net-section ultimate 59 5336 7-45 net-section ultimate 6 5336 2 bearing 6 5336 3 bearing 62 5336 4-45 net-section ultimate 63 5336 5 bearing 64 5336 5 bearing ultimate 65 5336 2 bearing ultimate 66 5336 3 bearing ultimate 67 5336 6 45 net-section ultimate \ Table 3 Maximum hole enlargement in a ree-bolt double-lap joint. Bolt, (in) Bolt 2, (in) Bolt 3 (in) Aluminum plate 4.68 x -5 3.53 x -5 3.74 x -5 Composite plate 3.933 x -5 5.723 x -5 6.9 x -5 Table 4 Spring stiffness values in a ree-bolt double-lap joint. Bolt (lb/in) Bolt 2 (lb/in) Bolt 3 (lb/in) Aluminum plate 4,338 53,58 57,99 45 29,7 22,589 28,862 Plies in composite plate 54,629 37,74 39,22-45 38,564 24,2 2,43 9 9,43 5,832 6,433 2

Fig. 8 Variation of ply loads rough e joint icness near bolt 3 in a ree-bolt joint. Fig. 9 Variation of bolt/plate displacement rough e joint icness near bolt 3 in a ree-bolt joint. 3

Load increment Table 5 Progressive ply failure for bolt in e ree-bolt double-lap joint. Ply Bolt load, Applied joint Ply orientation, (lb) load, (lb) number (degree) Failure mode 36 2242 9 shear-out 4 3673 22852 9 net-section 5 3673 22852 7 shear-out 7 37 238 9 net-section 8 37 238 5 shear-out 3747 233 3 shear-out 2 3784 23544 2 shear-out 4 3822 2378 8 shear-out 5 3822 2378 9 shear-out 7 386 247 2 shear-out 8 386 247 4 shear-out 9 386 247 6 shear-out 23 3977 24745 9 net-section ultimate 24 3977 24745 4-45 net-section 25 3977 24745 8-45 net-section 26 3977 24745 8-45 net-section ultimate 27 3977 24745 3-45 net-section 28 3977 24745 3-45 net-section ultimate 29 3977 24745 7-45 net-section 3 3977 24745 7-45 net-section ultimate 3 3977 24745 45 net-section 32 3977 24745 45 net-section ultimate 33 3977 24745 2 shear-out ultimate 34 3977 24745 4 shear-out ultimate 35 3977 24745 5 45 net-section 36 3977 24745 5 45 net-section ultimate 37 3977 24745 6 shear-out ultimate 38 3977 24745 8 shear-out ultimate 39 3977 24745 9 shear-out ultimate 4

Load increment Table 6 Progressive ply failure for bolt 2 in e ree-bolt double-lap joint. Ply Bolt load, Applied joint Ply orientation, (lb) load, (lb) number (degree) Failure mode 2 324 9 net-section 3 222 3255 9 net-section 7 245 586 9 net-section ultimate 22 253 5698 9 net-section ultimate 26 2589 674 8-45 net-section 29 264 6499 4-45 net-section 3 2668 6664 2 45 net-section 33 2694 683 7-45 net-section 35 272 6999 3-45 net-section 36 272 6999 6 45 net-section 4 284 754 45 net-section 4 284 754 5 45 net-section 46 297 8225 8-45 net-section ultimate 47 297 8225 3-45 net-section ultimate 48 297 8225 5 45 net-section ultimate 49 297 8225 45 net-section ultimate 5 297 8225 7-45 net-section ultimate 5 297 8225 4-45 net-section ultimate 52 297 8225 6 45 net-section ultimate 53 297 8225 2 shear-out 54 297 8225 3 shear-out 55 297 8225 5 shear-out 56 297 8225 7 shear-out 57 297 8225 9 shear-out 58 297 8225 2 45 net-section ultimate 59 297 8225 2 shear-out 6 297 8225 4 shear-out 6 297 8225 6 shear-out 62 297 8225 8 shear-out 63 297 8225 9 shear-out 69 366 955 2 shear-out ultimate 5

Load increment Table 7 Progressive ply failure for bolt 3 in e ree-bolt double-lap joint. Ply Bolt load, Applied joint Ply orientation, (lb) load, (lb) number (degree) Failure mode 232 2769 9 net-section 3 2325 2897 9 net-section 6 2372 356 9 shear-out 8 2395 3288 7 shear-out 249 342 5 shear-out 249 342 2 45 net-section 3 2443 3555 2 shear-out 4 2443 3555 3 shear-out 6 2468 369 6 shear-out 7 2468 369 8 shear-out 8 2468 369 9 shear-out 9 2468 369 6 45 net-section 2 2493 3827 2 shear-out 22 2493 3827 4 shear-out 24 257 3965 45 net-section 25 257 3965 5 45 net-section 26 257 3965 5 45 net-section ultimate 27 257 3965 45 net-section ultimate 28 257 3965 3-45 net-section 29 257 3965 7-45 net-section 3 257 3965 9 net-section ultimate 3 257 3965 9 net-section ultimate 32 257 3965 4-45 net-section 33 257 3965 4-45 net-section ultimate 34 257 3965 2 shear-out ultimate 35 257 3965 3-45 net-section ultimate 36 257 3965 4 shear-out ultimate 37 257 3965 6 shear-out ultimate 38 257 3965 7-45 net-section ultimate 39 257 3965 8 shear-out ultimate An initial joint load of P 32.5 lbs is applied to e composite laminate while e end of e aluminum plate is constrained. Due to e presence of symmetry in geometry and loading, only e results concerning bolts and 3 are presented. The maximum hole enlargement values associated wi ese bolt holes at were computed from e two-dimensional analysis are presented in Table 8, and e spring stiffness values for each ply are in Table 9. The rough-e-icness variation of e ply loads near bolt number is shown in Fig. 2. The corresponding bolt/hole deformations are depicted in Fig. 2. As observed in ese figures, e most pronounced deformation occurs in plies located along e plate interfaces. As presented in Table, e initial ply failure near bolt 3 occurs at a load level of 4,57 lbs, in ply number wi a 45 fiber orientation, in e netsection failure mode. The failure progresses wi e ±45 and 9 fiber orientations in e net-section mode, and furer continues wi e failure of plies wi fiber orientation in e bearing mode. Part of e laminate near bolt becomes unstable at load increment 5, corresponding to a load of 8,79 lbs, in ply 7 wi a -45 fiber orientation, in e netsection failure. At is load level, seven more failures occur in e composite laminate before e laminate is assumed to ultimately fail at load increment 7, at a load level of 2,599 lbs. Near bolt 3, e initial ply initial failure occurs at 8,26 lbs, in ply 2 wi fiber orientation, in e shear-out failure mode as presented in Table. The failure progresses wi plies of fiber orientation in shear-out mode. Part of e laminate near bolt 3 becomes unstable at load increment 29, corresponding to a load of 9,4 lbs, in ply 3 wi a -45 fiber orientation, in e net-section ultimate failure. At is load level, more failures occur in e composite laminate before e laminate 6

Table 8 Maximum hole enlargement in e fourbolt single-lap joint. Bolt, (in) Bolt 3, (in) Aluminum plate.444 x -4.369 x -4 Composite plate.8532 x -4.4378 x -4 Table 9 Spring stiffness values in e four-bolt single-lap joint. Bolt, (lb/in) Bolt 3, (lb/in) Aluminum plate 39,527 63,83 45 37,955 2,8 Plies in 5,8 38,542 composite -45 34,835 29,7 plate 9 8,747 6,56 Fig. 2 Variation of ply loads rough e joint icness near bolt 3 in e four-bolt joint. 7

Fig. 2 Variation of bolt/plate displacement rough e joint icness near bolt 3 in e four-bolt joint. Load increment Table Progressive ply failure for bolt in e four-bolt single-lap joint. Ply Bolt load, Applied joint Ply orientation, (lb) load, (lb) number (degree) Failure mode 3296 457 45 net-section 6 343 596 5 45 net-section 4 3678 685 6 45 net-section 5 3678 685 2 45 net-section 2 3827 6842 9 net-section 2 3827 6842 9 net-section 24 394 78 45 net-section ultimate 25 394 78 5 45 net-section ultimate 26 394 78 9 net-section ultimate 27 394 78 6 45 net-section ultimate 28 394 78 9 net-section ultimate 29 394 78 2 45 net-section ultimate 32 3984 7526 2 bearing 34 422 77 4 bearing 37 43 857 6 bearing 39 444 8238 8 bearing 4 444 8238 9 bearing 42 485 842 3-45 net-section 43 485 842 2 bearing 45 4227 864 3 bearing 46 4227 864 5 bearing 47 4227 864 7 bearing 48 4227 864 9 bearing 5 427 879 7-45 net-section 5 427 879 7-45 net-section ultimate 52 427 879 3-45 net-section ultimate 53 427 879 4-45 net-section 54 427 879 4-45 net-section ultimate 55 427 879 8-45 net-section 56 427 879 8-45 net-section ultimate 7 498 2599 2 bearing ultimate 8

Load increment Table Progressive ply failure for bolt 3 in e four-bolt single-lap joint. Ply Bolt load, Applied joint Ply orientation, (lb) load, (lb) number (degree) Failure mode 226 826 2 shear-out 4 235 8427 4 shear-out 6 2329 85 6 shear-out 8 2352 8596 8 shear-out 2375 8682 9 shear-out 2 2399 8769 2 shear-out 3 2399 8769 3 shear-out 4 2399 8769 5 shear-out 6 2423 8857 9 net-section 7 2423 8857 7 shear-out 8 2423 8857 9 shear-out 2 2447 8945 9 net-section 2 2447 8945 9 net-section ultimate 25 252 926 3-45 net-section 27 2547 938 7-45 net-section 29 2572 942 3-45 net-section ultimate 3 2572 942 7-45 net-section ultimate 3 2572 942 9 net-section ultimate 32 2572 942 4-45 net-section 33 2572 942 4-45 net-section ultimate 34 2572 942 45 net-section 35 2572 942 45 net-section ultimate 36 2572 942 2 shear-out ultimate 37 2572 942 4 shear-out ultimate 38 2572 942 5 45 net-section 39 2572 942 5 45 net-section ultimate 4 2572 942 6 shear-out ultimate 4 2572 942 8 shear-out ultimate is assumed to ultimately fail at load increment 4. Thus, e ultimate joint failure load occurs near bolt 3 at a load level of 9,42 lbs. Conclusions In is study, an approach to predict e streng of single- and double-lap bolted composites has been developed based on e rough-e-icness ply loads of e laminate in conjunction wi e average stress failure criterion. This approach utilizes e model of a beam on an elastic foundation to compute e corrected ply loads utilizing a two- dimensional stress analysis based on e complex potential and variational formulation. In e case of a one-bolt single-lap aluminum-to-composite joint, e joint streng prediction from e present approach is in acceptable agreement wi e experimental measurement published previously. This approach proves at e ply load distribution in a laminate is significantly influenced near e bolt by e bolt bending deformations. This distribution is dependent on e plate icness and laminate lay-up, and it is different for single- and double-lap bolted joints. References Kradinov, V., Barut, A., Madenci, E., and Ambur, D. R., Bolted Double-Lap Composite Joints Under Mechanical and Thermal Loading, International Journal of Solids and Structures, Vol. 38, 2, pp. 57-75. 2 Ramumar, R. L., and Saeer, E. S., Streng Analysis of Composite and Metallic Plates Bolted Togeer by a Single Fastener, Aircraft Division, Report AFWAL-TR-85-364, Norrop Corporation, Haworne, CA, August 985. 3 Whitney, J. M. and Nuismer, R. J., Stress Fracture Criteria for Laminated Composites Containing Stress Concentrations, Journal of Composite Materials, Vol. 8, 974, pp. 253-265. 9

4 Kradinov, V., Madenci, E., and Ambur, D. R., Analysis of Bolted Laminates of Varying Thicness and Lay-up wi Metallic Inserts, AIAA Paper 23-998, April 23. 5 Erisson, I., Baclund, J., and Moller, P., Design of Multiple-Row Bolted Composite Joints Under General In-Plane Loading, Composites Engineering, Vol.5, 995, pp. 5-68. 6 Xiong Y., An Analytical Meod for Failure Pre diction of Multi-Fastener Composite Joints, International Journal of Solids and Structures, Vol. 33, 995, pp. 4395-449. 7 Ghali, A., and Neville, A. M., Structural Analysis: A Unified Classical and Matrix Approach, Chapman and Hall, New Yor, 978. Appendix In e laminate, e strain energy of e l bolt, which is defined by a uniform cross-section, of inertia, I l, and Young s and shear moduli, E l and G (, respectively, can be expressed as A l, moment B 2 q b q i K l l l i T i i (A) where q i l represents e vector of nodal deflections and rotations for e i beam element and K is e number of nodes in e bolt discretization. The stiffness matrix for a two-node Timosheno beam element is given by Ghali and Neville 7 as b i l + α ( il ) ( l ) ( l ) 2E I symmetric 3 ( il ) ( h ) ( l ) ( l ) ( l ) ( l ) 6E I ( il ) E I ( 4 + α 2 ) il il h h ( l ) ( l ) ( l ) ( l ) ( l ) ( l ) 2E I 6E I 2E I 3 2 3 ( il ) ( il ) ( il ) ( h ) ( h ) ( h ) 6E I ( il ) E I 6E I ( il ) E I 2 ( 2 α ) 2 ( 4+ α i l ) il h ( il ) ( h ) h h l l l l l l l l ( il ) (A2) ( i ) ( i ) 2 where α l ( h l ) G l A l /2E l I l c in which c represents e shear correction factor. Rearranging e right-hand side of Eq. (A) such at e matrices are suitable for static condensation of e internal nodal rotations leads to where B ( l ) 2 T b b,, φ T q q, φ l l b b, φ, φ, φφ (A3) { φ L φ,,,2,3,,, } T K K K q { φ φ L φ } T φ K,,2,3,( ) (A4) 2

b b b,, φ, φφ ( l) ( l) ( l) b b b,,3,2 ( l) ( l) ( l) b b b,3,33,23 ( l) ( l) ( l) (2 l) (2 l) b b b + b b,2,23,22,,2 O O O O b b + b b b b b b b b b ( l) b,4 ( l) b,34 ( l) (2 l) (2 l) b + b b,24,3,4 (2 l) (2 l) (3 l) (3 l) b b + b b,23,24,3,4 O O O (( K 3) l) (( K 3) l) (( K 2) l) (( K 2) l) b b + b b,23,24,3,4 l l b b + b (( K ) l) b,23 (( K ) ) b l,34 ( l) (2 l) (2 l) b + b b,44,33,34 (2 l) (2 l) (3 l) (3 l) b b + b b,34,44,33,34 O O O (( K 3) l) (( K 3) l) (( K 2) l) (( K 2) l) b b + b b,34,44,33,34 K l K l b b + b (( K 2) l) (( K 2) l) (( K ) l) (( K ) l) (( K ) l),2,22,,2,4 (( K ) l) (( K ) l) (( K ) l),2,22,24 (( K ) l) (( K ) l) (( K ) l),4,24,44 (( K 2) ) (( K 2) ) (( K ) l),23,24,3 (( 2) ) (( 2) ) (( K ) l),34,44,33 (A5) (A6) (A7) Since e internal nodes are not subjected to external moments, e first variation of e strain energy wi respect to e vector q ( l ) vanishes, resulting in e moment equilibrium equations as, φ b b q T, φ +, φφ, φ (A8) Solving for ( l ) q, φ in e Eq. (A8) and substituting into Eq. (A3), after rearranging e terms, leads to B 2 b T (A9) ( l ) where e matrix b is defined as b b b b b 2 T φ,,,, φφ φ (A) The strain energy of e translational and rotational spring elements can be expressed as S 2 T (A) in which ( l ) ( l ) is defined as ( l ) is e same as wi an additional degree of freedom representing e rigid base for e springs, and 2

, h,,,2,2 O, N, N, N, N N h,,,2 N, N, i, i, n (A2) where e last column and row appear due to degrees of freedom associated wi e rigid base. Since e rigid base is fixed, is column and row are eliminated from e matrix in Eq. (A2). Thus, e total strain energy of e beam on an elastic foundation can be written as or in which ( l ) is defined by U B S l + ( ) U 2 b + T ( l ) ( l ) ( l ) ( l ) ( l ) (A3a) (A3b), h,,2 O (A4), N, N n, In e case of a single-lap bolted joint, e bolt nut is not present in e top plate,, while e bolt head is not present in e bottom plate, 2; us, in e top plate and ( l ) in e bottom plate. In e absence of, n 2, h bo head and nut in e middle plate of a double-lap bolted joint, in addition to e values of for e top and bottom plates, 2, h 2, n in e middle plate., n 3, h 22

The total potential energy of e l bolt can be expressed as where and K π U W 2 K F, b + ( l ) ( l ) T b + 2 2 M P node K F P 2 node K + M (A5) for a single-lap joint, and { φ L φ 2 + + } T K M K M b + K b + 2 2 and b + 3 3, { M ( l ) P ( l ) P node K F M K + P + P P node K + φ 2 3 ( l ) P3 K + M + M node M node L φ K φ } T 2 K + M K + M K + M + P K + M + P (A6) (A7) for a double-lap joint. The variables K, M, and P define e number of nodes in e beam discretization for each plate. According to e bilinear behavior of e ply loads shown in Fig., e coefficients of e matrix become ˆ [ ˆ l l α, where α.] for initially damaged plies and zero for totally damaged plies. Also, for initially l i, i, i, damaged plies, e values of ( ˆ IN l, l i, ) l i, j are added to e corresponding locations in e vector of externally applied loads, F. Appling e principle of minimum potential energy and forcing e first variation of e total potential to vanish, δπ, leads to e system of equilibrium equations 23

K F (A9) In e finite element formulation of e problem, additional constraint equations are introduced to ensure e continuity of e rotational degree of freedom between e adjacent section of e beam φ φ K K + φ φ K K + φ φ K + M K + M + for a single lap joint for a double lap joint (A2) These constraint conditions are enforced by adding an additional row and column wi all but two elements set to zero in e matrix K. Two nonzero elements are set to and - in e locations corresponding to e nodal rotations; a zero value is added in e right-hand side. 24