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

Transcription

1 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

2 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

3 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

4 (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

5 (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

6 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

7 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

8 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

9 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) 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, L 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 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

10 () 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 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.

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

12 Load increment Table 2 Progressive ply failure in one-bolt single-lap joint. Ply Applied joint Ply orientation, load, (lb) number (degree) Failure mode net-section net-section net-section ultimate net-section ultimate net-section net-section net-section bearing bearing net-section bearing bearing bearing net-section net-section net-section net-section net-section ultimate net-section ultimate net-section ultimate net-section ultimate bearing bearing net-section ultimate bearing bearing ultimate bearing ultimate bearing ultimate 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 x x -5 Composite plate x x 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, ,7 22,589 28,862 Plies in composite plate 54,629 37,74 39, ,564 24,2 2,43 9 9,43 5,832 6,433 2

13 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

14 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 shear-out net-section shear-out net-section shear-out shear-out shear-out shear-out shear-out shear-out shear-out shear-out net-section ultimate net-section net-section net-section ultimate net-section net-section ultimate net-section net-section ultimate net-section net-section ultimate shear-out ultimate shear-out ultimate net-section net-section ultimate shear-out ultimate shear-out ultimate shear-out ultimate 4

15 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 net-section net-section net-section ultimate net-section ultimate net-section net-section net-section net-section net-section net-section net-section net-section net-section ultimate net-section ultimate net-section ultimate net-section ultimate net-section ultimate net-section ultimate net-section ultimate shear-out shear-out shear-out shear-out shear-out net-section ultimate shear-out shear-out shear-out shear-out shear-out shear-out ultimate 5

16 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 net-section net-section shear-out shear-out shear-out net-section shear-out shear-out shear-out shear-out shear-out net-section shear-out shear-out net-section net-section net-section ultimate net-section ultimate net-section net-section net-section ultimate net-section ultimate net-section net-section ultimate shear-out ultimate net-section ultimate shear-out ultimate shear-out ultimate net-section ultimate 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

17 Table 8 Maximum hole enlargement in e fourbolt single-lap joint. Bolt, (in) Bolt 3, (in) Aluminum plate.444 x x -4 Composite plate.8532 x 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, ,955 2,8 Plies in 5,8 38,542 composite ,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

18 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 net-section net-section net-section net-section net-section net-section net-section ultimate net-section ultimate net-section ultimate net-section ultimate net-section ultimate net-section ultimate bearing bearing bearing bearing bearing net-section bearing bearing bearing bearing bearing net-section net-section ultimate net-section ultimate net-section net-section ultimate net-section net-section ultimate bearing ultimate 8

19 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 shear-out shear-out shear-out shear-out shear-out shear-out shear-out shear-out net-section shear-out shear-out net-section net-section ultimate net-section net-section net-section ultimate net-section ultimate net-section ultimate net-section net-section ultimate net-section net-section ultimate shear-out ultimate shear-out ultimate net-section net-section ultimate shear-out ultimate 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 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 , Norrop Corporation, Haworne, CA, August Whitney, J. M. and Nuismer, R. J., Stress Fracture Criteria for Laminated Composites Containing Stress Concentrations, Journal of Composite Materials, Vol. 8, 974, pp

20 4 Kradinov, V., Madenci, E., and Ambur, D. R., Analysis of Bolted Laminates of Varying Thicness and Lay-up wi Metallic Inserts, AIAA Paper , April 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 Xiong Y., An Analytical Meod for Failure Pre diction of Multi-Fastener Composite Joints, International Journal of Solids and Structures, Vol. 33, 995, pp 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 ( 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

21 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

22 , 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

23 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 M P node K F P 2 node K + M (A5) for a single-lap joint, and { φ L φ } T K M K M b + K b 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

24 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