Analysis of bolted–bonded composite single-lap joints under combined in-plane and transverse loading

Composite Structures 88 (2009) 579–594

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Analysis of bolted–bonded composite single-lap joints under combined in-plane and transverse loading A. Barut, E. Madenci * The University of Arizona, Tucson, AZ 85721, United States

a r t i c l e

i n f o

Article history: Available online 14 June 2008 Keywords: Lap joint Bolt Bond Composite

a b s t r a c t A semi-analytical solution method is developed for stress analysis of single-lap hybrid (bolted/bonded) joints of composite laminates under in-plane as well as lateral loading. The laminate and bolt displacements are based on the Mindlin and Timoshenko beam theories, respectively. For the adhesive, the displacement field is expressed in terms of those of laminates by using the shear-lag model. The derivation of the governing equations of equilibrium of the joint is based on the virtual work principle, where the kinematics of each laminate are approximated by local and global functions and the bolt kinematics is assumed in terms of cubic Hermitian polynomials. The capability of the present approach is justified by validation and demonstration problems, including the analysis of bolted and bonded joints and hybrid joints with and without considering a disbond between the adhesive and laminates. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction Bolted, bonded, and hybrid bolted/bonded joints are threedimensional in nature. Although the finite element method (FEM) is capable of addressing such joint configurations, it requires considerable computer resources, especially in the presence of multiple bolts. The presence of unknown contact regions between the bolts and laminates and the small length scale of the adhesive thickness require a fine mesh to achieve a reliable prediction of the stress field. Therefore, identification of critical design parameters and optimization of the joint strength have become a computational challenge with the finite element method. In order to alleviate extensive computations in the case of a bolted or bonded joint analysis, semi-analytical methods exist to predict the stress field in both bolted and bonded lap joints. While the tools developed for the bolted joint analyses are limited to two-dimensional stress field in the laminates, the semi-analytical solution tools for bonded joint analysis include three-dimensional deformations but are limited to thin plate theory. Although there have been a substantial number of experimental, analytical, and numerical investigations on the stress analysis of bolted and bonded composite laminates, previous studies concerning the analysis of bolted/bonded lap joints in the open literature are limited [1–6] and, in all of these analysis, the FEM was the method of choice to determine the stress fields in the bolted– bonded joints. As mentioned earlier, the efficiency of the FEA is drastically reduced, as the adhesive requires a highly refined mesh * Corresponding author. Tel.: +1 520 621 6113; fax: +1 520 621 8191. E-mail address: [email protected] (E. Madenci). 0263-8223/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2008.06.003

in order to keep the proper aspect ratio between the elements in the laminates and adhesive. Furthermore, the area near the estimated contact region between the bolts and laminates must be properly meshed in order to keep track of the contact region, which requires an iterative solution of the governing equations in which the global system matrix is repeatedly calculated. In this case, a three-dimensional finite element analysis of the entire domain becomes computationally demanding. Therefore, it would be beneficial to have an efficient specialpurpose analysis method that can be used to conduct extensive parametric studies in a timely manner and at relatively low computational costs. For this purpose, the aim of the present study is to develop a semi-analytical solution method that permits the determination of point-wise variation of displacement and stress components and the bolt load distribution in single-lap bolted/ bonded joints of composite laminates under general loading. The derivation of the governing equations of equilibrium of the joint is based on the principle of virtual work, where the displacement fields in the laminates are represented by local and global functions that are not required to satisfy the kinematic boundary conditions directly and the displacements of bolts that are assumed based on the three-dimensional Timoshenko beam theory. For the adhesive, the displacement field is expressed in terms of those of laminates by using the shear-lag model to approximate both shearing and peeling deformations; hence, no assumed displacements are necessary for the adhesive. Although the present formulation is based on small-deflection theory and utilizes linear material laws for the laminates, bolts, and adhesive, the solution of the resulting equilibrium equation requires an iterative procedure due to the presence of unknown contact regions.

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2. Problem description The hybrid lap joint consists of composite laminates, an adhesive bond layer, and a bolt in the middle, as shown in Fig. 1. As shown in this figure, the bolt is clamped to the laminates by the bolt heads from the top and bottom of the lap joint. The laminates can be subjected to combined surface and boundary loadings. As shown in Fig. 2, the domains of the laminates and adhesive layer are denoted by AðpÞ ðp ¼ 1; 2Þ and AðaÞ , respectively, and are located on the mid-surface of both the laminates and adhesive. The bolt is idealized as a beam. Hence, its domain is represented by the centroidal line, ‘ðbÞ . The boundaries of laminates along which displacement constraints and external loads are applied are denoted by ‘ðpÞ ðp ¼ 1; 2Þ. The laminates have rectangular geometry whose length, width, ðpÞ and thickness are specified by LðpÞ ; W ðpÞ , and 2h ðp ¼ 1; 2Þ, respectively. The adhesive bond represents the overlap area between the laminates, as shown in Fig. 2. The overlap region is also

Fig. 1. Geometric description of the bolted–bonded joint under general loading.

described by a rectangular prism whose dimensions are specified ðaÞ as the length, LðaÞ , width, W ðaÞ , and thickness, 2h , of the adhesive layer. As for the bolt, the geometric dimensions include the length and radius of the bolt, LðbÞ and r ðbÞ , as well as the outer radius of the ðbÞ clamp-up region, r cl , over which the bolt heads apply pressure to the laminates, as shown in Fig. 3. The clamp-up region, Acl , is conðbÞ fined to the region defined by r ðbÞ 6 r 6 r cl . As shown in this figure, there also exists a clearance (gap) between the bolt and hole boundary of the laminates. The size of the clearance is specified by d0 , and is much smaller than that of the bolt radius (i.e., d0 =r ðbÞ  1). The effect of clearance appears in the contact conditions between the bolt and hole boundary. The laminates can be made of layered composite laminates, where each layer in a laminate has transversely isotropic material properties defined by elastic moduli, EL and ET , shear modulus, GLT , and Poisson’s ratio, mLT , in which L and T refer to the fiber and transverse (perpendicular to fiber) directions, respectively. Also, each layer can have arbitrary thickness, t k , and ply angle, hk , with respect to the global x-axis, as shown in Fig. 2. However, the adhesive is made of isotropic material, whose elastic modulus and Poisson’s ratio are specified as EðaÞ and mðaÞ , respectively. The bolt is made of an isotropic material, with Young’s and shear moduli specified as EðbÞ and GðbÞ , respectively. Also, it is idealized as a beam with a circular cross-section whose axial, flexural, and shear stiffnesses are specified as EðbÞ AðbÞ ; EðbÞ IðbÞ , and GðbÞ AðbÞ , respectively, with AðbÞ and IðbÞ representing the area and first moment of inertia of the cross-section of the bolt. In order to create clamp-up forces, the bolt is pre-stressed (or pre-stretched) prior to loading. The pre-stress conditions are mathematically invoked into the total potential energy expression through the bolt’s constitutive relations described in the subsequent section. As shown in Fig. 1, two types of external loads can be applied on ðpÞ the joint: (1) distributed surface pressure load,  p0 ðp ¼ 1; 2Þ, and (2) line loads along the edges. Along the boundary of the laminates, the line loads can be of any type of resultant tractions, denoted as ðpÞ  ðpÞ N k , and moments,  mk ðk ¼ x; y; z; p ¼ 1; 2Þ. The positive directions of these resultant tractions and moments are shown in Fig. 1. The kinematic boundary conditions are applied in an indirect way by employing spring support mechanisms, as illustrated in

Fig. 2. Idealized description of the bolted–bonded joint.

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the laminate, the kinematic boundary conditions are enforced via the strain energy of these elastic supports. As illustrated in Fig. 4 (bottom), the second type of boundary condition, which is very common in experimental setups, concerns the application of concentrated forces through a rigid load fixture mounted onto the laminate edge. This type of boundary condition is idealized by the spring-rigid end bar mechanism shown in Fig. 4 (bottom). The extensional springs are placed between the laminate boundary, ‘ðpÞ , and the rigid end bar, DðpÞ , which is subjected to conðpÞ ðpÞ centrated normal,  P Dx , and shear,  P Dy , forces. After loading, the riand gid bars experience unknown uniform displacements of DðpÞ x DðpÞ y , which are determined from the solution of the equilibrium equations. The strain energy of this support mechanism is obtained from the stretching of the normal and shear springs having spring ðpÞ ðpÞ stiffness properties of kDx and kDy , respectively. Furthermore, the potential energy of the applied concentrated forces acting on the loading fixture is calculated as part of the total potential energy expression. The problem posed in the present study is the determination of stresses in the laminates, adhesive, and bolt when the bolt is pretensioned to produce a clamp-up force effect while the joint is subjected to external loading and prescribed displacement constraints. Fig. 3. Detailed view of the bolt and hole boundary.

3. Formulation Fig. 4. As shown in this figure, two possible types of kinematic boundary conditions can be applied along the same edge, i.e., ‘ðpÞ ðp ¼ 1; 2Þ. The first type of kinematic boundary condition is ðpÞ the prescribed displacements,  uk ðk ¼ x; y; zÞ, and slopes,  ðpÞ hj ðj ¼ x; yÞ, through springs attached to the laminate edge, as shown in Fig. 4 (top). In this figure, the extensional and rotational ðpÞ ðpÞ spring attachments are denoted as kj and khj ðj ¼ x; y; z; p ¼ 1; 2Þ, respectively. By applying appropriate spring constants to the extensional and rotational springs, which can be several orders of magnitude higher than the in-plane and bending stiffnesses of

The solution method is based on the principle of minimum potential energy. Requiring the first variation of the total potential energy to vanish leads to the governing equations. The resulting equations govern the stress fields in the laminates, the adhesive, and the bolt deformation. 3.1. Laminate kinematics and strain energy Based on the Mindlin plate theory, in which the transverse shear strains are assumed uniform through the thickness, the displacement field at a generic point along the thickness of the laminate is expressed in terms of the mid-surface displacements and independent slopes in the form

9 ðpÞ ðpÞ > U ðpÞ x ðx; y; zÞ ¼ ux ðx; yÞ  ðz  zp Þhx ðx; yÞ > = ðpÞ

ðpÞ U ðpÞ y ðx; y; zÞ ¼ uy ðx; yÞ  ðz  zp Þhy ðx; yÞ

U ðpÞ z ðx; y; zÞ

¼

ðpÞ uz ðx; yÞ

> > ;

ðp ¼ 1; 2Þ

ð1a—cÞ

U ðpÞ x ;

ðpÞ where U ðpÞ represent, respectively, the Cartesian y , and U z components of the displacements at a generic point in the plate ðpÞ ðpÞ ðpÞ in the x-, y-, and z-directions, and ux ; uy , and uz represent the Cartesian mid-surface displacement components in the x-, y-, and z-directions, respectively. Also, the mid-surface slope components ðpÞ in the x- and y-directions are denoted by hðpÞ x and hy , respectively. Using the strain–displacement relations, the strain components of laminates at a generic point can be expressed as

ðpÞ ðpÞ EðpÞ x ¼ ex þ ðz  zp Þjx ;

ðpÞ ðpÞ EðpÞ y ¼ ey þ ðz  zp Þjy

ðp ¼ 1; 2Þ ð2a; bÞ

ðpÞ ðpÞ CðpÞ xy ¼ cxy þ ðz  zp Þjxy ;

ðpÞ CðpÞ xz ¼ cxz ;

ðpÞ CðpÞ ðp ¼ 1; 2Þ yz ¼ cyz

ð2c—eÞ ðpÞ ðpÞ where EðpÞ x ; Ey , and Cxy denote the in-plane normal and shear

strains and CðpÞ and CðpÞ denote the transverse shear strain xz yz components. Defined on the mid-surface of the laminate, the ðpÞ

ðpÞ

ðpÞ

ðpÞ

ðpÞ

resultant in-plane ðex ; ey ; cxy Þ and transverse shear ðcxz ; cyz Þ strain components and curvature ðj are expressed as Fig. 4. Elastic support conditions to facilitate the imposition of boundary conditions.

ðpÞ eðpÞ x ¼ ux;x ;

ðpÞ eðpÞ y ¼ uy;y

ðpÞ x ;

ðp ¼ 1; 2Þ

j

ðpÞ y Þ

and twist ðj

ðpÞ xy Þ

components

ð3a; bÞ

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ðpÞ ðpÞ ðpÞ ðpÞ ðpÞ ðpÞ cðpÞ cðpÞ cðpÞ ðp ¼ 1; 2Þ xy ¼ ux;y þ uy;x ; xz ¼ uz;x  hx ; yz ¼ uz;y  hy

ð3c—eÞ ðpÞ ðpÞ ðpÞ ðpÞ jðpÞ jðpÞ jðpÞ ðp ¼ 1; 2Þ x ¼ hx;x ; y ¼ hy;y ; xy ¼ hx;y  hy;x

cðaÞ jz ¼

ð2Þ

¼

ð3f—hÞ

8 9ðpÞ 2 3ðpÞ 8 9ðpÞ A B 0 > > = = 6 7 ¼ 4B D 05 j M > > ; : > : > c; 0 0 G Q

ð4Þ

ðp ¼ 1; 2Þ

where the subvectors e; j, and c contain the components of the inplane strain, curvature, and twist, as well as the transverse shear strain resultants as ðpÞ ðpÞ eT ¼ feðpÞ x ; ey ; cxy g;

The corresponding in-plane stress resultant components, bending moments, and transverse shear stress resultants are contained in the vectors N, M, and Q as

MT ¼ fM x ; M y ; M xy g;

ð1Þ

U j j

z¼zð2Þ hð2Þ

ð1Þ

U z j

2h

9 > > > > > ð2Þ ð2Þ ð1Þ ð1Þ > > þ h h þ h h Þ=

z¼zð1Þ þhð2Þ

j

j

ðaÞ

ð2Þ

ð1Þ

¼ 2h1ðaÞ ðuz  uz Þ ðaÞ

ðj ¼ x; yÞ

> > > > > > > ;

z¼zð2Þ þhð2Þ

ð10a; bÞ

ðaÞ

where cjz and ez denote the transverse shear and transverse normal strain resultants in the adhesive, respectively. In accordance ðaÞ with the shear-lag model, the shear stresses, rjz ðj ¼ x; yÞ, and ðaÞ transverse normal stress, rz , in the adhesive are related to the ðaÞ transverse shear and normal strain resultants, cjz ðj ¼ x; yÞ and ðaÞ ez , through

9

8 > <

3 8

2

9

ðaÞ ðaÞ ðaÞ rz > E > = = < ez > 6 7 cxz rxz ¼4 G 5 > > > ; : :c > ryz ; G yz

ð11Þ

In matrix form, Eq. (11) can be rewritten as

ðpÞ ðpÞ ðpÞ jT ¼ fjðpÞ cT ¼ fcðpÞ x ; jy ; jxy g; xz ; cyz g

ð5a—cÞ

NT ¼ fNx ; Ny ; Nxy g;

Uz j

z¼zð2Þ hð2Þ

2hðaÞ ð2Þ ð1Þ ðuj  uj

¼ 2h1ðaÞ ðaÞ ez

In accordance with the Yang–Norris–Stavsky theory, the resultant stresses in the laminates are expressed in terms of the resultant strains and curvatures as

ð2Þ

Uj j

Q T ¼ fQ xz ; Q yz g

rðaÞ ¼ EðaÞ eðaÞ where T

2

E

where ðN x ; N y ; Nxy Þ and ðMx ; M y ; Mxy Þ are the in-plane stress resultants and bending moments, respectively, and ðQ xz ; Q yz Þ represent the transverse shear stress resultants, all along the x- and y-directions, respectively. The resultant stress–resultant strain relation in Eq. (4) is represented in compact form as

T

ðaÞ ðaÞ ðaÞ ðaÞ rðaÞ ¼ frðaÞ cðaÞ ¼ fcðaÞ z ; rxz ; ryz g; z ; cxz ; cyz g;

ðaÞ

ð6a—cÞ

ð12Þ

6 ¼4

3ðaÞ

E

7 5

G

ð13a—cÞ

G Finally, the strain energy of the adhesive bond, U ðaÞ , is expressed as

U ðaÞ ¼

1 2

Z A

T

eðaÞ EðaÞ eðaÞ dA

ð14Þ

ðaÞ

3.3. Bolt kinematics and strain energy

sðpÞ ¼ CðpÞ eðpÞ

ð7Þ

with T

T

T

T

sðpÞ ¼ fNðpÞ ; MðpÞ ; Q ðpÞ g; 2 3ðpÞ A B 0 6 7 CðpÞ ¼ 4 B D 0 5 0

0

T

T

T

T

eðpÞ ¼ feðpÞ ; jðpÞ ; cðpÞ g; ð8a; b; cÞ

G

Using Eq. (7), the strain energy of the laminate can be expressed in matrix form as

U ðpÞ ¼

1 2

Z

T

eðpÞ CðpÞ eðpÞ dA

AðpÞ

ð9Þ

3.2. Adhesive kinematics and strain energy The adhesive bond between the laminates of the lap joint is an isotropic material whose elastic modulus, EðaÞ , may be considerably smaller than that of the laminates. The strain energy of the adhesive due to its transverse normal (peel) and transverse shear deformations is much higher than that due to its in-plane deformations. Therefore, the effect of in-plane deformations of the adhesive bond is neglected simply because of its low resistance against in-plane deformations. The adhesive primarily transfers the loading from one laminate to the other through its transverse normal and shear stiffness properties. The behavior of the adhesive therefore can be idealized through a shear-lag model applicable to both transverse shear and transverse normal deformations in the form

The bolt is considered to be flexible through its length, and its mechanical behavior is modeled based on the Timoshenko beam theory, which is analogous to the Mindlin plate theory in that it accounts for uniform transverse shear deformations in both transverse directions of the bolt. Based on the sign convention shown in Fig. 3, the displacement of a generic point in the bolt can be expressed in terms of the displacements and slopes defined along the neutral axis of the bolt as ðbÞ U ðbÞ x ðx; y; zÞ ¼ ux ðzÞ;

ðbÞ U ðbÞ y ðx; y; zÞ ¼ uy ðzÞ

ð15a; bÞ

ðbÞ ðbÞ ðbÞ ðbÞ ðbÞ U ðbÞ z ðx; y; zÞ ¼ uz ðzÞ þ ðx  x Þhx þ ðy  y Þhy

ð15cÞ

ðbÞ

where U k ðk ¼ x; y; zÞ are the Cartesian components of the displacements defined at an arbitrary point on the circular cross-secðbÞ ðbÞ tion of the bolts, and uk ðk ¼ x; y; zÞ and hj ðj ¼ x; yÞ denote the displacement and independent slope components defined at a point along the longitudinal (neutral) axis of the bolt, respectively. In accordance with the Timoshenko beam theory, the strains that create strain energy in the bolt are the axial strain, EðbÞ z , and transverse ðbÞ shear strain components, Czj ðj ¼ x; yÞ. They can be expressed in terms of the derivatives of the axial displacement and slope components of the bolt as ðbÞ ðbÞ ðbÞ ðbÞ ðbÞ EðbÞ z ¼ ez þ ðx  x Þjx þ ðy  y Þjy ;

ðbÞ CðbÞ ðj ¼ x; yÞ zj ¼ cj

ð16a; bÞ ðbÞ ez ;

ðbÞ j ,

ðbÞ j

where j and c ðj ¼ x; yÞ represent, respectively, the resultant strain, curvature, and transverse shear strain components defined as

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A. Barut, E. Madenci / Composite Structures 88 (2009) 579–594 ðbÞ ðbÞ ðbÞ ðbÞ jðbÞ ¼ hj;z ; cj ¼ uz;j þ hj ða ¼ x; yÞ j

ezðbÞ ¼ uðbÞ z;z ;

ð17a—cÞ

The bolt is made of an isotropic material and has a circular crosssection. The resultant forces associated with the strain resultants defined in Eq. (17) are the axial force component, N ðbÞ x , the bending ðbÞ moments, M ðbÞ x and M y , and the transverse shear force components, ðbÞ Q ðbÞ x and Q y . These resultant forces and moments act along the axis of the bolt; their positive directions are shown in Fig. 5. The bolt can be pre-stretched in order to produce a clamp-up ðbÞ force, N x0 , prior to loading, as shown in Fig. 5. In order to take into account the initial stresses generated in the bolt due to the clampup force, its resultant force–resultant strain relations are expressed in a general form as

9ðbÞ 2 8 Nz > EA > > > > > > > 6 > > M EI > 6 = < x> 6 6 My ¼6 > > > > 6 > > > Qx > 4 > > > > ; : Qy

EI

9ðbÞ 3ðbÞ 8 9ðbÞ 8  ez > Nz0 > > > > > > > > > > > > > 7 >  > > jx > Mx0 > > > > > 7 > = = < < 7  7 j 7 > y > þ > M y0 > > > >c > > 7 > 2 > Q x0 > k GA > > 5 > x > > > > > > > > ; ; : > : 2 cy Q y0 k GA ð18Þ

2

where k denotes the shear correction factor for a circular cross-section. In Eq. (18), the initial forces and moments are denoted by ðbÞ ðbÞ ðbÞ ðbÞ  ðbÞ  N z0 ; Mx0 ;  My0 ;  Q x0 , and  Q y0 , and they are treated as known. In this analysis, all of the initial forces, except for the axial force ðbÞ component,  N x0 , are set to zero. Hence, Eq. (18) can be rewritten in matrix form as ðbÞ

sðbÞ ¼ CðbÞ eðbÞ þ  s0

ð19Þ

3.4. Bolt–laminate contact conditions In the deformed configuration shown in Fig. 6, vertical contact between the bolt and hole boundary is full through the laminate thickness. However, partial contact occurs along the circumference of the bolt–hole. Also, the contact between the bolt heads and laminates is fully extended to the size of the bolt head because the bolt is initially pre-tensioned and it applies clamp-up pressure to the laminates. 3.4.1. Bolt–hole boundary contact condition Initially, there exists a clearance of magnitude d0 between the bolt surface and the hole boundary, as depicted in Fig. 7a. Upon loading, the laminate moves relative to the bolt, and once the relative displacement between the hole boundary and bolt reaches d0 , the first contact between the bolt and hole boundary occurs at point A, as shown in Fig. 7b. Until this point, no contact stresses develop, i.e., the resultant of the contact stresses, Rc , is zero. An increase in loading, however, causes more relative displacement to occur, thus resulting in the extension of the contact region along both sides of point A along the circumference of the hole boundary as shown in Fig. 7c; thus, contact stresses develop. As shown in Fig. 7c, the contact region is defined by the angles u1 and u2 . The initial contact point, A, is expected to lie in the middle of these contact angles, i.e., u0 ¼ ðu1 þ u2 Þ=2. Based on the contact angles, u1 and u2 , while satisfying the continuity of radial displacements along the contact region and including the presence of clearance, the constraint condition between the bolt and laminate displacements can be established as ðpÞ U ðbÞ r  U r  qr ¼ 0

where

ðpÞ

T

sðbÞ ¼ fNz ; M x ; My ; Q x ; Q y gðbÞ ; T

T  ðbÞ s0

¼ f Nz0 ; 0; 0; 0; 0gðbÞ 2

eðbÞ ¼ fez ; jx ; jy ; cx ; cy gðbÞ ;

ð20a; bÞ

2

CðbÞ ¼ Diag½EA; EI; EI; k G; k GðbÞ ð20c; dÞ

Furthermore, using Eq. (19), the strain energy of the bolt is expressed in matrix form as

U ðbÞ ¼

1 2

Z ‘

T

eðbÞ CðbÞ eðbÞ dz þ

Z

T

ðbÞ

eðbÞ s0 dz

ð21Þ

‘

on zðpÞ  h

ð22Þ ðpÞ

6 z 6 zðpÞ þ h

and u1 6 u 6 u2 , where

9 ðbÞ ðbÞ U ðbÞ r ¼ U x cosðuÞ þ U y sinðuÞ > =

ðpÞ ðpÞ u1 6 u 6 u2 U ðpÞ r ¼ U x cosðuÞ þ U y sinðuÞ > ; q ¼ q0 ð1  cosðu  u0 ÞÞ

ð23a—cÞ

Substituting for the laminate and bolt displacements from Eq. (23), this contact condition along the hole boundary can be rewritten as ðbÞ ðpÞ ðpÞ ðpÞ DU ðb;pÞ ¼ uðbÞ x cosðuÞ þ uy sinðuÞ  ðux  ðz  z Þhx Þ cosðuÞ c ðpÞ ðpÞ þ ðuðpÞ y  ðz  z Þhy Þ sinðuÞ  2q0 ð1  cosðuÞÞ ¼ 0 ð24Þ ðpÞ

ðpÞ

on zðpÞ  h 6 z 6 zðpÞ þ h and u1 6 u 6 u2 . This constraint condition then can be invoked into the total potential energy expression using a Lagrange multiplier function representing the constraint stresses (or contact stresses) in the form

Fig. 5. Idealized model of the contact regions between the bolt and laminates.

Fig. 6. Contact regions between the bolt and laminates in a deformed configuration.

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A. Barut, E. Madenci / Composite Structures 88 (2009) 579–594

analysis, the spring mechanism shown in Fig. 5 is adopted to model the interaction between the bolt heads and laminates. As shown in Fig. 5, the vertical extensional springs having a substantially high stiffness coefficient are attached between the bolt head and laminate boundary. The relative displacements between the bolt heads and laminates basically yield the amount of spring extension (or contraction) at the interfaces. The amount of spring extension/contraction along the interfaces at the bottom and top of the joint is expressed in terms of the bolt and laminate displacement components as

DU ðb;pÞ ¼ U zðbÞ jz¼0  U ð1Þ z ; 0 Sðb;pÞ , r

DU ðb;pÞ z

ð2Þ DU ðb;pÞ ¼ U ðbÞ z jz¼LðbÞ  U z L

ð26a; bÞ

ðbÞ

on where ðz ¼ 0; L Þ denotes the relative displacements between the bolt heads and the laminates measured at the bottom ðz ¼ 0Þ and top ðz ¼ LÞ of the joint. In Eq. (26), the vertical displacements of the bolt heads, i.e., U ðbÞ z jz¼0;LðbÞ , are simply obtained from the extrapolation of the bolt displacement field along the interface (contact) region. Substituting for the displacement fields of both the laminates and bolt from Eqs. (1) and (15) yields ðbÞ ðbÞ ðbÞ ðbÞ ð1Þ DU ðb;pÞ ¼ ðuðbÞ z þ ðx  x Þhx þ ðy  y Þhy Þz¼0  uz 0 ðbÞ ðbÞ ðbÞ ðbÞ ð2Þ DU ðb;pÞ ¼ ðuðbÞ z þ ðx  x Þhx þ ðy  y Þhy Þz¼LðbÞ  uz L

ð27a; bÞ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi on r ðbÞ 6 ðx  xðbÞ Þ2 þ ðy  yðbÞ Þ2 6 r cl . The extensional springs between the bolt heads and laminates, shown in Fig. 5, have uniform spring stiffness of kcl , which is at least three to four orders of magnitude higher than that of both the laminates and bolt. Applying high extensional spring stiffness ðb;pÞ ðb;pÞ and DU L to become zero, thus satisfying the forces both DU 0 contact conditions between the bolt heads and the laminates. The bolt head–laminate contact conditions are invoked into the total potential energy by evaluating the strain energy of the extensional springs along their interface as

Xðb;pÞ ¼ cl

1 2

¼

1 2

Z Acl

Z

Acl

D

kcl U ðb;pÞ zp dA ðb;pÞ

kL

ðbÞ ðbÞ ðbÞ ðbÞ ðpÞ ½ðuðbÞ z þ ðx  x Þhx þ ðy  y Þhy Þz¼zp  uz dA

ð28Þ with z1 ¼ 0 and z2 ¼ LðbÞ , and Acl denotes the interface between the bolt heads and the laminates, as shown in Fig. 5. 3.5. Boundary conditions

Fig. 7. Bolt and hole boundary contact region (a) before loading, (b) at the onset of contact after loading, and (c) after load exertion through the bolt.

V ðb;pÞ ¼ c

Z

zðpÞ þhðpÞ

zðpÞ hðpÞ

Z

u2

u1

kDU ðb;pÞ r ðbÞ du dz  0 c

ð25Þ

where k represents the unknown contact stress distribution along the bolt–hole boundary contact region. This equation is often referred to as the zero potential energy of the constraint forces. 3.4.2. Bolt head–laminate contact condition Full contact between the (rigid) bolt heads and laminates exists due to the presence of clamp-up forces generated by pre-tensioning of the bolt. As a result, transverse displacement of the laminates must be the same as the vertical (z-) displacement of the bolt heads on the contact surface, as illustrated in Fig. 6. In this

3.5.1. Prescribed boundary displacements and slopes The prescribed boundary conditions along the specified edges, shown in Fig. 4 (top), are controlled through elastic springs. Using appropriate spring stiffness values for each extensional and rotational spring, the boundary displacements and boundary slopes of the laminates can be fixed to zero, or to a prescribed value. In order to incorporate the prescribed boundary conditions in the equilibrium equations, the strain energy of the springs is included in the total potential energy expression. The stretch or contraction of the extensional springs is defined as ðpÞ  ðpÞ DuðpÞ k ¼ uk  uk

ðk ¼ x; y; z; p ¼ 1; 2Þ

ð29aÞ

ðpÞ

on ‘ðpÞ , where  uk is the prescribed boundary displacement. The slope of the rotational springs is defined as ðpÞ  ðpÞ DhðpÞ k ¼ hk  hk ðpÞ

 ðpÞ hk

ðk ¼ x; y; p ¼ 1; 2Þ

ð29bÞ

on ‘ , where are the prescribed slopes along the laminate boundary, ‘ðpÞ . Using Eq. (29), the strain energy of the elastic springs along the laminate boundaries is expressed as

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A. Barut, E. Madenci / Composite Structures 88 (2009) 579–594

XðpÞ u ¼

1 X 1X ðpÞ ðpÞ ðpÞ ðpÞ kuj ðuj   uj Þ2 þ khj ðhj   hj Þ2 2 j¼x;y;z 2 j¼x;y

ð30Þ

3.5.2. Concentrated force through a rigid load fixture This type of boundary and loading condition is commonly used in experiments; that is, a concentrated load is applied through a rigid load fixture mounted on the laminate edge. The model shown in Fig. 4 (bottom) is adopted to simulate such boundary conditions. In this model, the rigid end bar represents the loading fixture. The strain energy of the springs between the laminate and rigid end bar and the potential energy of the concentrated load applied through the rigid end bar are added to the total potential energy expression. The strain energy due to the stretching of extensional springs between the laminate boundary and the rigid end bar is evaluated from ðpÞ D

X

1 X ðpÞ ðpÞ ðpÞ ¼ k ðD  uj Þ2 2 j¼x;y Dj j

equilibrium equations. Therefore, a semi-analytical solution method is developed by adopting the Rayleigh–Ritz procedure. 4. Solution method 4.1. Displacement representation The solution method begins with the selection of the assumed functions representing the displacements and slope fields of the laminates and bolt, as well as the assumed solution form for the unknown Lagrange multiplier function representing the contact (constraint) stresses between the bolt and laminates. The unknown ðpÞ ðpÞ fields of the laminates, i.e., uk and hk ðk ¼ 1; 2; 3; l ¼ 1; 2; p ¼ 1; 2Þ, are expressed in terms of the superposition of assumed local and global displacement fields in the form ðpÞ

ui

ðpÞ

ðpÞ

 ; i þ u ¼u i

ðpÞ

hl

¼ hl þ hl ðpÞ

ðpÞ

ði ¼ x; y; z; l ¼ x; y; p ¼ 1; 2Þ

ð31Þ

ð36a; bÞ

where Dj ðj ¼ x; yÞ is the unknown uniform displacements of the rigid end bar in the x- and y-directions, respectively, as shown in Fig. 4 (bottom). Also, the potential energy of the concentrated normal and shear forces applied to the rigid end bar is expressed as

where the single and double bars represent, respectively, the local and global fields. In Eq. (36), the local displacement/slope fields are assumed in the form of a local double series in terms of polar coordinates, whose origin is attached at the center of the hole boundary in the form

ðpÞ

ðpÞ

VD ¼

X

 ðpÞ ðpÞ PDj Dj

ð32Þ

j¼x;y

 ðpÞ u ¼ l

Nr X

a ðlÞðpÞ ln r T j ðhÞ þ 0j

j¼1

Nt Nr X X

r i T j ðhÞ ðl ¼ x; y; p ¼ 1; 2Þ a ðlÞðpÞ ij

i¼N r j¼1

ðpÞ

where P Dj and Dj ðj ¼ x; yÞ are concentrated loads acting on the load fixture and rigid-body displacements of the loading fixture in the x- and y-directions. 3.5.3. Externally applied loads The external pressure applied in the vertical (z-) direction and external distributed line loads acting along the ends of the laminates are considered, as shown in Fig. 1. The potential energy of these loads is expressed in the form

V ðpÞ pr

¼

V ðpÞ r ¼

Z AðpÞ

 ðpÞ ðpÞ p0 uz dA

Z ‘ðpÞ

on A

ðpÞ

ð33Þ

 ðpÞ ðpÞ   ðpÞ  ðpÞ   Nx uðpÞ þ N u þ N u þ m h þ m h d‘ y y z z x x y y x

ð34Þ

3.6. Total potential energy expression The total potential energy that governs the equilibrium of the bonded–bolted joint is expressed by adding the strain energy of the laminates, Eq. (9), adhesive, Eq. (14), and bolt, Eq. (21); the extensional springs along the bolt head–laminate interface, Eq. (28); the extensional and rotational springs along the laminate boundaries, Eq. (30); and the potential energies of the external, Eqs. (31)–(34), and constraint forces, Eq. (25), in the form 2 X

U ðpÞ þ U ðaÞ þ U ðbÞ þ

2 X

p¼1



2 X p¼1

XðpÞ u þ

p¼1

V ðb;pÞ  c

2 X p¼1

V ðpÞ pr 

2 X p¼1

2 X

XðpÞ D þ

p¼1

V ðpÞ r 

2 X

2 X

Xðb;pÞ cl

p¼1 ðpÞ

VD

 ðpÞ u z ¼

Nr X

rðln r  1ÞT j ðhÞ þ a ðzÞðpÞ 0j

j¼0

i¼N r j¼0 i–0;1

hðpÞ ¼ l

 a ðzÞðpÞ 1j ln r T j ðhÞ

j¼0 Nt a Nr  ðzÞðpÞ X X ij

þ

Nr X

Ns X

iþ1

r iþ1 T j ðhÞ ðp ¼ 1; 2Þ

ðlÞðpÞ ln r T j ðhÞ þ b 0j

j¼1

Nr Ns X X

ð37bÞ

ðlÞðpÞr i T j ðhÞ ðl ¼ x; y; p ¼ 1; 2Þ b ij

i¼N r j¼1

ð37cÞ

where  N k ðk ¼ x; y; zÞ and  mk ðk ¼ x; yÞ are the externally applied resultant tractions and moments, respectively. Their positive directions are illustrated in Fig. 1.

p¼

ð37aÞ

ð35Þ

with

r ¼ r=rðbÞ ;

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx  xðbÞ Þ2 þ ðy  yðbÞ Þ2

ð38a; bÞ

The series representation in the radial direction is an expanded form of the power series that additionally includes a logarithmic term and negative powers of the radial variable, r , while the series representation in the circumferential direction is based on equally spaced periodical B-spline functions, T j ðhÞ. The details of these periodical B-spline functions, T j ðhÞ, can be found in Appendix A. In Eq.  ðpÞ (31), the unknown coefficients of the local displacement, u k , and ðpÞ ðlÞðpÞ ðk ¼  ðlÞðpÞ and b slope, hl , fields are represented by a ij ij x; y; z; l ¼ x; y; p ¼ 1; 2Þ, respectively. The logarithmic terms in the local functions are used to ensure a 1=r variation in all stress resultants and moment resultants of the laminates. For out-of-plane dis ðpÞ r terms used for the placement component, u z , the integrals of the  in-plane and slope components are employed for compatibility in Eq. (37b). As for the global displacement/slope fields, a double series in the form of Bernstein polynomials is employed in both plane coordinates, i.e.,

p¼1

The equations of equilibrium are obtained by enforcing the first variation of the total potential energy to vanish, i.e., dp ¼ 0. Due to the presence of the contact region between the bolt and the hole boundary, as well as the finite geometry of the laminates, it is impossible to construct a closed-form solution for the resulting

r ¼

ðpÞ

 ðpÞ ¼ u k hðpÞ ¼ l

ðpÞ

My Mx X X i¼0

j¼0

ðpÞ

ðpÞ

i¼0

j¼0

My Mx X X

ðpÞ ðpÞ a ðkÞðpÞ Bi ðtx ÞBj ðty Þ ðk ¼ x; y; z; p ¼ 1; 2Þ ij

ð39aÞ

ðlÞðpÞ BðpÞ ðt ÞBðpÞ ðt Þ ðl ¼ x; y; z; p ¼ 1; 2Þ b x y ij i j

ð39bÞ

586

A. Barut, E. Madenci / Composite Structures 88 (2009) 579–594 ðpÞ

where 0 6 tl 6 1 and M l denotes the degree and extent of the series used for each direction and laminate. In Eq. (39), the unknown ðpÞ  ðpÞ coefficients of the global displacement, u k , and slope, hl , fields ðlÞðpÞ ðlÞðpÞ   and bij ðk ¼ x; y; z; l ¼ x; y; p ¼ 1; 2Þ, are represented by aij respectively. The Bernstein polynomials are defined as ðpÞ

Ml !

ðpÞ

Bk ðt l Þ ¼

ðpÞ ðMl

ðpÞ

 kÞ!k!

ð1  t l ÞðMl

kÞ k tl

ð40Þ

Each Bernstein term attains a maximum value at a point in the doðpÞ main (i.e., the point at which dBk =dt l ¼ 0Þ, and they monotonically reduce to zero at the end points (with the exception of the first and last terms, which are equal to unity at these points). Hence, each Bernstein term can be referred to as a local function that is effective around its extremum point. Therefore, the superposition of all Bernstein terms provides a complete representation of the global fields. As for the bolt, cubic Hermitian polynomials for transverse disðbÞ ðbÞ placements, ux and uy , complete quadratic polynomials for the slope components, hxðbÞ and hðbÞ y , and a linear polynomial for the axðbÞ ial displacement component, uz , are initially assumed. By applying the constraint of uniform transverse shearing forces along the length of the bolt, the reduced representation for the displacement and slope fields in the bolt is achieved as

9  ðbÞ > > ¼ 1 þ þ 5z  2L þ 3L2 hlð0Þ > 6 > > >     > > ðbÞ ðbÞ > z z2 2z3 2z 2z2 4z3 þ  6  2L þ 3L2 hlðLÞ þ  3 þ L  3L2 hlðmÞ > > > > > =   ðbÞ ðbÞ z z ðbÞ ðl ¼ x; yÞ uz ¼ 1  L uzð0Þ þ L uzðLÞ > > >     > 2 2 > ðbÞ ðbÞ ðbÞ 3z > hl ¼ 2zL2  Lþ1 > hlð0Þ þ 2zL2  Lz hlðLÞ > > > >   > > ðbÞ > 4z2 4z ; þ  þ h ðbÞ ul



 z

ðbÞ ulð0Þ L

L2

z ðbÞ u L lðLÞ



3z2

2z3

lðmÞ

L

ð41a—cÞ ðbÞ ukð0Þ

ðbÞ ulðLÞ

where and ðk ¼ x; y; zÞ are the displacements of the bolt evaluated at the bottom and top intersections with the bolt heads, ðbÞ ðbÞ as shown in Fig. 3. Similarly, hlð0Þ and hlðLÞ ðl ¼ x; yÞ denote the slopes of the bolt evaluated at the bottom and top intersection points with ðbÞ the bolt. Also, the slope components, hlðmÞ ðl ¼ x; yÞ, are evaluated at the mid-point ðz ¼ LðbÞ Þ along the axial line of the bolt. The unknown Lagrange multiplier functions, representing the contact stress distribution, along the contact region between the bolt and hole boundary are considered uniform through the thickness of the laminate and vary in the circumferential direction such that the contact stresses become zero at the beginning and end ðpÞ ðpÞ points (u1 and u2 ) of the contact region. Hence, the Lagrange multiplier functions, kðpÞ , become functions of the contact angle, u, and are assumed in terms of sine series in the circumferential direction in the form ðpÞ

k

ðpÞ

¼ k ðuÞ ¼

Lc X n¼1

kðpÞ n

2 X 1 ðpÞT ðpÞ ðpÞ 1 ðbÞT ðbÞ ðbÞ q K q þ q K q 2 2 p¼1 ( )" ð1;1Þ #( ) T Kð1;2Þ Ka qð1Þ 1 qð1Þ a þ T 2 qð2ÞT qð2Þ Kð1;2Þ Kð2;2Þ a a ( ) " # ( ) T ðb;bÞ ðb;1Þ K0 K0 qðbÞ 1 qðbÞ þ ðb;1ÞT ð1;1Þ 2 qð1ÞT qð1Þ K0 K0 ( ) " # ( ) T ðb;bÞ ðb;2Þ KL KL qðbÞ 1 qðbÞ þ ðb;2ÞT ð2;2Þ 2 qð1ÞT qð1Þ K K

P¼

u  uðpÞ 1 sin np ðpÞ uðpÞ 2  u1

! ð42Þ

where the constant coefficients kðpÞ n denote the unknown Lagrange multipliers to be determined as part of the solution.

4.2. Equations of equilibrium

L

2 X

T

 qðpÞ  PðpÞ pr 

DTp  PðDp Þ 

p¼1

2 X

T

kðb;pÞ ðGðb;pÞ qðbÞ

p¼1

þ GðpÞ qðpÞ   Pðb;pÞ q Þ

ð43Þ

The total potential energy expression, Eq. (43), can be further rearranged in more compact form by defining an overall solution vector, q, in the form T

T

T

T

T

qT ¼ fqð1Þ ; qð2Þ ; qðbÞ ; DT1 ; DT2 ; kðb;1Þ ; kðb;2Þ g

ð44Þ

resulting in

1 2

P ¼ qT Hq  qT P

ð45Þ

where

2

Hð1Þ 6 ð1;2ÞT 6H 6 6 ð1;bÞT 6H 6 T H¼6 6 Hð1;D1 Þ 6 6 0 6 6 4 Gð1Þ

P¼

Hð1;2Þ

Hð1;bÞ

Hð1;D1 Þ

0T

Gð1Þ

Hð2Þ

Hð2;bÞ

0T

Hð2;D2 Þ

0

HðbÞ

0T

0T

0

HðD1 Þ

0

0

0

0 0T

HðD2 Þ 0T

0 0

0T

0T

0

Hð2;bÞ

T

0 ð2;D2 ÞT

H

0T

Gðb;1Þ

Gð2Þ Gðb;2Þ 0T 8  ð1Þ  ð1Þ  ð1Þ 9 P þ Ppr þ Pu > > > > > > r > > >  ð2Þ  ð2Þ >  ð2Þ > > P þ P þ P > > r pr u > > > > > > > >  ðbÞ > > > > Pcl = < > > > > > > > > > > > > > :



PðD1 Þ



PðD2 Þ



Pðb;1Þ q



Pðb;2Þ q

T

ðbÞT

G0

0

3

T 7 Gð2Þ 7 7 ðbÞT 7 GL 7 7 7 0 7; 7 0 7 7 7 0 5

0

ð46a; bÞ

> > > > > > > > > > > > > ;

with ð1;1Þ

ð1;1Þ þ Kð1Þ ; u þ KD ð2;2Þ ð2;2Þ ð2Þ þ KL þ Ku þ KD ; H ¼K þ ðb;bÞ ðb;bÞ HðbÞ ¼ KðbÞ þ K0 þ KL ; Hð1;2Þ ¼ Kð1;2Þ ; a ð1;bÞ ð2;bÞ ð1;bÞ ð2;bÞ ð1;D1 Þ H ¼ K0 ; H ¼ KL ; H ¼ Kð1;D1 Þ ; ð2;D2 Þ ð2;D2 Þ ðD 1 Þ ðD 1 ;D 1 Þ ðD 2 Þ ðD2 ;D2 Þ

Hð1Þ ¼ Kð1Þ þ Kað1;1Þ þ K0 ð2Þ

H

ð2Þ

¼K

Kað2;2Þ

;

H

¼K

;

H

¼K

ð47a—jÞ

Enforcing the first variation of the total potential energy expression, Eq. (43), to vanish yields the matrix equations of equilibrium of the bolted–bonded joint in the form

Hq ¼ P After expressing all of the strain and potential energies arising from bolted–bonded joint components in matrix form, as presented in detail in Appendix B, their substitution into Eq. (35) leads to the matrix representation of the total potential energy as

L

2 2 2 X X T T 1 ðpÞT ðpÞ ðpÞ X ðbÞT  ðbÞ þ qðpÞ  PðpÞ qðpÞ  PðpÞ Pcl q Ku q  r þq u  2 p¼1 p¼1 p¼1

ð48Þ

In Eq. (48), the system matrix contains submatrices HðpÞ ; HðbÞ , and HðDp Þ ðp ¼ 1; 2Þ, which represent the self-stiffness matrices associated with the unknown generalized coordinates of the laminates,

A. Barut, E. Madenci / Composite Structures 88 (2009) 579–594

bolt, and the rigid load fixture, respectively. Furthermore, the offdiagonal submatrices, Hðk;jÞ ; Hðb;pÞ , and Hðp;Dp Þ ðk ¼ 1; 2; j ¼ 1; 2; p ¼ 1; 2Þ, denote the coupling stiffness terms between the laminates due to the adhesive layer, the laminates and the bolt due to the clamp-up condition, and the laminates and the rigid load fixture due to uniform end displacement conditions. Since the kinematics of the adhesive is represented by the laminates, based on the shear-lag theory, their stiffness terms are automatically added to the self- and off-diagonal stiffness terms of the laminates. Similarly, the stiffness of the elastic spring supports and those of the springs between the laminates and bolt heads are directly added to the laminate stiffness matrices. In the system matrix expression in Eq. (46a), the direct imposition of the constraints of the contact condition between the bolt and laminate hole boundary is represented by the coefficient matrices Gðb;1Þ ; Gð1Þ ; Gðb;2Þ , and Gð2Þ . In the overall right-hand-side load vector, P, in Eq. (46b), the first two components are the load vectors acting on the laminates arising from the direct application of boundary tractions and moðpÞ ments, PðpÞ u , the distributed load, Ppr , as well as the loading due to known kinematic boundary conditions of the laminates. The ðbÞ next term, Pcl , contains the forces arising from initial stretching of the bolt. These forces are associated with the unknowns of the beam; however, they are transferred to the other components of the joint due to coupling terms (effects) in the system matrix, H. ðpÞ The next two terms,  PD1 , contain the components of the applied concentrated forces acting on the rigid load fixture. Hence, they are associated with the rigid end bar displacements. Finally, the ðb;pÞ last two terms, Pq , are the loading calculated as a result of the presence of clearance between the bolt and hole boundary. 5. Numerical results The present approach is first validated against previously developed and verified models for bolted and bonded lap joint analyses under uniaxial tension, as shown in Fig. 8a and b, respectively. The same exterior geometry, material, and boundary conditions are used for both cases. As shown in Fig. 8, the length and width of the laminates are specified as L1 ¼ L2 ¼ 84 mm and W 1 ¼

587

W 2 ¼ W ¼ 24 mm. The length and width of the overlap region, where the laminates are either bolted or bonded, are specified as La ¼ 24 mm and W a ¼ W. Also, each laminate thickness is specified as h1 ¼ h2 ¼ 2 mm. Both laminates of the lap joint are quasi-isotropic and have identical ply material properties. Each ply is transversely isotropic and their properties in the principal material axes are specified as EL ¼ 180 GPa; ET ¼ 10:3 GPa; GLT ¼ 7:17 GPa, and mLT ¼ 0:28, where L and T refer to the fiber and transverse directions of the principal material axes. The stacking sequence of the quasi-isotropic laminates is specified as ½0=90=  45=452s . As shown in Fig. 8, the laminates are restrained along their unjoined edges, ‘ðpÞ , against the transverse displacement, i.e.,  ðpÞ uz ¼ 0, and rotation about the y-axis, i.e.,  hðpÞ x ¼ 0. While the horizontal displacement of the second laminate along its un-joined edge is restrained, a concentrated force of 24 kN is applied through a load fixture (rigid end bar) placed along the un-joined edge of the first plate. In the case of a bolted joint, the bolt is located at the center, as shown in Fig. 8a, and the bolt length is measured between the bottom and top surfaces of the overlap region and is specified as Lb ¼ h1 þ h2 ¼ 4 mm. As shown in this figure, the bolt radius is specified as r b ¼ 2 mm. Although no clamp-up pressure is considered, bolt heads are still modeled in order to provide appropriate contact between the laminates through the bolt heads and also to prevent a mathematically overdetermined (singular) system matrix. The radius of the bolt heads at the bottom and top of the bolt are specified as r cl ¼ 3 mm. The bolt is made of steel, with Young’s modulus of E ¼ 193 GPa and Poisson’s ratio of m ¼ 0:3. The present analysis predictions are compared against a previously developed bolted joint analysis method by Kradinov et al. [7]. The previous method computes only the in-plane behavior of the joint and does not take into account bending due to coupling effects. A comparison of the radial (contact) and hoop stresses around the bolt in the laminates is shown in Fig. 9. Although both models have distinct features, they both yield almost identical stresses around the bolt hole. This is expected because both models contain only a single bolt, which transmits the entire applied load from one plate to the other with or without the presence of

Fig. 8. Geometric description of (a) bolted and (b) bonded lap joints.

588

A. Barut, E. Madenci / Composite Structures 88 (2009) 579–594

Fig. 9. Comparison of (a) radial, N r , and (b) hoop, N h , resultant stresses for the bolted joint analysis.

bending deformations in the laminates. Therefore, the resultant of the contact stresses around the bolt–hole in the laminates must yield values identical to the bolt load. In the case of a bonded joint, the laminates are joined through an adhesive bond, as illustrated in Fig. 8b. The bond thickness is ðaÞ h ¼ 0:125 mm. The adhesive is isotropic, with a shear modulus of G ¼ 0:414 GPa and a Poisson’s ratio of m ¼ 0:34. The present analysis predictions are compared against a previously developed bonded joint analysis method by Oterkus et al. [8]. A comparison of the adhesive stresses along the horizontal centerline of the overlap region is shown in Fig. 10. As shown in this figure, both models yield almost identical shear stresses in the adhesive. Furthermore, both models capture the steep peel and shear stress variations almost identically. Note that the variation of the stresses obtained by the present approach has a tendency to oscillate around the actual results, as observed in the peel stress variation in Fig. 10. This is basically due to the use of a globally defined set of functions, i.e., Bernstein

Fig. 10. Comparison of shear, rxz , and peel, rzz , stresses in the adhesive of a bonded joint.

polynomials, in the present model. In the previous model, B-spline functions were utilized and the accuracy could easily be controlled by using control (referred to as ‘‘knot”) point definitions. Typical deformed configurations for bolted and bonded joint analyses are shown in Fig. 11. After validating the present approach separately for bolted and bonded joint analyses, the applicability of the present approach is demonstrated by considering a bolted–bonded composite lap joint subjected to the loading cases of (a) uniform end stretching and (b) combined uniform end stretching and transverse distributed loading, all with and without inducing a bolt clamp-up force, as depicted in Fig. 12. Except for the adhesive thickness of ðaÞ h ¼ 0:2 mm, the geometric and material configurations, and the kinematic boundary conditions of the joint in this problem are identical to those considered in the validation problems. As shown in Fig. 12, the magnitudes of axial and transverse distributed loadings are specified as P0 ¼ 24 kN and p0 ¼ 0:5 MPa, respectively. Both in the presence and absence of the transverse distributed loading and clamp-up forces, the solution of the hybrid joint reveals that the centrally located bolt does not carry (or transfer) any significant load because of the small relative displacement between the laminates, which is insufficient to establish the contact between the laminate hole boundaries and the bolt. Therefore, the present hybrid joint problem is investigated considering only the contact between the laminates and the bolt heads due to clampup forces. However, the case of zero clamp-up force is also considered in order to understand the effect of clamp-up forces induced by the pre-stretching of the bolt. The adhesive stresses along the horizontal centerline of the overlap region for clamp-up forces of N 0 ¼ 0, 2, and 4 kN in both uniaxial and combined loading cases are shown in Fig. 13. In both cases, it can be seen that all the peel stresses near the hole boundary become compressive as the clamp-up force is applied. However, the shear stresses in the adhesive do not change with the clamp-up force. This is mainly because the present shear-lag model adopted for the adhesive excludes the coupling between transverse expansion and shear deformations. Away from the cutout, both the peel and shear stresses are unaffected by the presence of the clamp-up force. As given in Tables 1 and 2, it is observed that the initial clamp-up forces are reduced by 40% after equilibrium is reached. This is mainly due to the contraction in the adhesive bond, which results in a reduction in bolt stretching after loading. Comparing the deformed configurations shown in Fig. 14, the uniaxial loading, load case (a), results in asymmetric bending deformations as a result of secondary bending moments caused by the eccentric in-plane loading. In the case of combined loading, load case (b), the present approach captures the non-symmetric bending deformations as a result of the presence of transverse distributed loading, which yields higher bending deformations on the left side ðx < 0Þ of the joint than on the right side ðx > 0Þ. The effect of asymmetric and non-symmetric bending deformations of the joint can also be observed from the comparison of the adhesive stress distributions in Fig. 13. In the case of uniaxial end

Fig. 11. Deformed configurations of (a) bolted and (b) bonded single-lap joints.

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Fig. 12. Description of a hybrid lap joint with a pre-tensioned bolt clamped along the edges against bending and subjected to (a) uniaxial and (b) combined uniaxial and vertical distributed loadings.

they vary non-symmetrically in the case of combined loading, shown in Fig. 13b. It is observed that the shear and peel stresses attain higher values along the left edge of the overlap area than along the right edge in the case of combined loading. Comparisons of Fig. 14a and b and Tables 1 and 2 further reveal that the presence of transverse distributed loading has a negligible effect on the variation of the adhesive stresses around the clampup area as well as the resulting clamp-up forces on the bolt.

Table 1 Pre-tension load applied to the bolt before uniaxial loading and the resulting clampup force measured after loading Pre-tensile force on bolt (N)

Clamp-up force on hybrid joint (N)

0 2000 4000

0 1564 3066

Table 2 The pre-tension load applied to the bolt before combined axial loading and pressure and the resulting clamp-up force measured after loading Pre-tensile force on bolt (N)

Clamp-up force on hybrid joint (N)

0 2000 4000

0 1571 3073

Fig. 13. Adhesive stresses with varying clamp-up loading under (a) uniaxial loading and (b) combined uniaxial and vertical distributed loadings.

stretching, shown in Fig. 13a, both the shear and peel stresses are symmetric along the horizontal centerline of the joint. However,

Fig. 14. Deformed configurations of the hybrid joint under (a) uniaxial and (b) combined uniaxial and distributed loadings.

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A. Barut, E. Madenci / Composite Structures 88 (2009) 579–594

Fig. 15. Geometric description of a hybrid joint with partial adhesive debonding.

As mentioned earlier, the main function of the bolt, other than reducing the peel stresses, is to prevent or delay failure initiation and progress in the joint due to adhesive–laminate disbonding. When partial disbonding occurs in the overlap region due to either initial defects or local peel failure in the adhesive, the applied load increases the relative in-plane displacements between the laminates, thus causing contact between the bolt and the laminates to develop. In this case, the applied load is partially transferred by the bolt. In order to understand the load transfer mechanism through the bolt in a hybrid joint, a partially disbonded hybrid lap joint subjected to uniaxial end stretching, as shown in Fig. 15, is considered. The width of the entire lap joint and the length of the overlap region are specified as W ¼ 48 mm and LðaÞ ¼ 12 mm, respectively. Also, a length parameter, W d , is introduced that represents the length of the disbonded area, as shown in Fig. 15. The disbond length, W p , is increased gradually between W p ¼ 25, 28, 32, 36, 40, and 44 mm. The hybrid joint analysis is then solved to calculate the load transferred by the bolt using the present approach where a fixed bolt–hole contact condition between the bolt and the laminates is considered. The fixed contact

Fig. 16. Variation of bolt load increase as a function of debond length.

regions are specified from 90° to 270° for the first (lower) and from 90° to 90° for the second (upper) laminates. Fig. 16 shows the variation of the total load transferred by the bolt as a function of the debond width, W d . Note that the analysis starts with a minimum debond length of W d ¼ 12:5 mm. This is the smallest debond length where the bolt starts to transfer some load. As seen in this figure, the bolt load increases with increasing debond length, as expected. It is also noted that the load taken by the bolt rapidly increases as the debond length approaches the horizontal overlap edges. As seen in Fig. 16, the load transfer capacity of the bolt reaches slightly over 20% of the overall joint load when W d is 44 mm (just 2 mm from the horizontal overlap edges). The bolt contact stresses as a function of increase in debond length is presented in Fig. 17. In all cases, the contact stress distributions contain two local peaks, one of which is slightly higher than the other. This is due to the effect of the bending and twisting

Fig. 17. Radial (a) and hoop (b) stresses around laminate holes for selected values of the debond length parameter, W d .

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coupling behavior of the quasi-isotropic laminates. It is observed that the contact stresses in the middle are smaller than the local peak stresses in all cases. This is due to the vertical contraction of the bolt–holes trying to push away the mid-point of the contact region. This behavior tends to diminish as the debond length approaches the horizontal overlap edges, as shown in Fig. 17. 6. Summary and conclusions In this study, a semi-analytical method was developed for the coupled in-plane and bending analysis of composite bonded– bolted single-lap hybrid joints. The bending is mainly due to the force couples created by the lap geometry. The laminates are joined through an adhesive bond layer along the overlap region between the laminates and a pre-tensioned bolt placed right at the center of the overlap region. The validation against bolted and bonded joint analyses showed the ability of the present approach in tackling both problems separately. The comparison of the results proved the robustness and accuracy of the present approach in capturing the correct response of the contact stresses in the bolted joint analysis and adhesive shear and peel stresses in the bonded joint analysis. In the hybrid bonded–bolted joint analysis, it is not surprising that most of the load is transferred through the adhesive, even though it has low elastic modulus as compared to the bolt. This is due to the fact that the adhesive has a large area and small thickness, thus yielding substantially high adhesive shear stiffness that does not allow enough relative displacements between the laminates to exert forces on the bolt in opposite directions. Thus, the force exerted by the bolt (in case contact occurs) would be negligibly small. For this reason, the following particular cases were investigated in this study: (1) the joint has no initial defects, no debonding occurs, and all loads are transferred by the adhesive bond (no contact between the bolt and the plates), and (2) the adhesive bond is partially debonded, resulting in the bolt transferring some of the load. In the first case, the effects of varying clamp-up forces and the uniaxial and combined loadings are investigated. The present analysis results show that the clamp-up force induced by the bolt stretching applies compression to the laminates through the bolt heads, causing compressive peeling stresses around the bolt head regions. Furthermore, the presence of the transverse loading significantly changes the distributions of shear and peel stresses in the adhesive, whereas it has a negligible effect on the clamp-up forces. In the second case, the effect of the increasing debond length on the lap joint is investigated. It is observed that no bolt load transfer occurs up to a certain debond length. Once the debond length continues to increase beyond this (critical) debond length, the bolt starts to take some of the load. The bolt load transfer rapidly increases and takes the entire load as full debonding occurs in the adhesive. Appendix A The B-spline functions, T i ðh; t; KÞ, employed in Eq. (37) are defined recursively in the form

T i ðh; tðpÞ ; KÞ ¼

ðh  t i Þ T i ðh; t; K  1Þ ðt iþK1  t i Þ þ

ðt iþK  hÞ T iþ1 ðh; t; K  1Þ ðt iþK1  t iþ1 Þ

t ¼ ft 0 ; t1 ; . . . ; tNB ; t NB þ1 ; t NB þ2 ; . . . ; t NB þ2K g ¼ fh0 ; h1 ; . . . ; hNB ; hNB þ1 ¼ h0 ; hNB þ2 ¼ h1 ; . . . ; hNB þK ¼ hK1 g

where N B is the number of knot points chosen in the circumferential direction around the hole boundary. These knot points are equally spaced in intervals hk ¼ ðk  1Þ2p=N B . Also, the last K components of the knot vector are forced to be identical to its first K components to ensure periodicity of the B-spline functions (i.e., closed B-spline functions) between 0 and 2p. In this study, fourth-degree B-spline functions (i.e.,K ¼ 5) are assumed in both directions in order to ensure continuity of both displacement and stress fields in the laminates. Finally, the relationship between Nt and the number of knot points, NB , is established as

Nt ¼ NB  2

ðA3Þ

Appendix B In order to systematically derive each of the terms in the total potential energy expression, Eq. (35), the assumed forms of the displacement and slope fields given in Eqs. (36) and (41) and the assumed form of the contact stresses defined in Eq. (42) are expressed in matrix form. The assumed solution forms of the displacement and slope fields in the laminates, defined in Eq. (36), are represented in matrix form as ðpÞT

ðpÞ

uk ¼ Vk qðpÞ ; ðpÞT

Vk

ðpÞT

¼ Vhl qðpÞ

ðk ¼ x; y; z; l ¼ x; yÞ

ðB1a; bÞ

n o n o ðpÞT ðpÞT ðpÞT ðpÞT ðpÞT ; Vhl ¼ Vhl ; Vhl ¼ Vk ; Vk

ðB2a; bÞ

and

n T o T  ðpÞT  ðpÞ ; q qðpÞ ¼ q

ðB3Þ

in which the generalized unknown vectors for local and global dis , are defined as  and q placements, q n o T T T T T T T T  ðpÞ  ðpÞ  ðpÞ  ðpÞ  ðpÞ  ðpÞ  ðpÞ  ðpÞ ¼ q q N r 0 ; qN r 1 ; .. .; qN r N t ; qðN r 1Þ0 ; .. .; qðN r 1ÞNt ; .. .; qN r 0 ; .. .; qN r N t

ðB4aÞ

 ðpÞT ¼ q



T

T

T

T

T

T

T

 ðpÞ ; q  ðpÞ ; . . . ; q  ðpÞ ðpÞ ; q  ðpÞ ; . . . ; q  ðpÞ ðpÞ ; . . . ; q  ðpÞðpÞ ; . . . ; q  ðpÞðpÞ q 00 01 10 0My

1My



ðpÞ M x My

Mx 0

ðB4bÞ  and q  contains where each subvector in vectors q T

ðxÞðpÞ ; b ðyÞðpÞ g  ðpÞ  ðxÞðpÞ  ðyÞðpÞ  ðzÞðpÞ ¼ fa ;a ;a ;b q ij ij ij ij ij ij

ðB5aÞ

T  ðpÞ q ij

ðB5bÞ

ðxÞðpÞ f  ij ;

¼ a

 ðyÞðpÞ ; ij

a

ðxÞðpÞ ; b ðyÞðpÞ g  ðzÞðpÞ ; b ij ij ij

a

In Eq. (B2), the explicit expressions for the vectors of known coeffiðpÞ  ðpÞ cient functions of the local and global functions, ðVk ; V hl Þ and ðpÞ ðpÞ ðVk ; Vhl Þ, with k ¼ x; y; z and l ¼ x; y, are explicitly defined in Appendix C. ðbÞ ðbÞ The bolt displacement and slope fields, uk and hl , with k ¼ x; y; z and l ¼ x; y, defined in Eq. (41) are also expressed in similar vector notations as ðbÞ

with i ¼ 1; 2; . . . ; N t and K > 1. The variable tj represents the components of the knot vector, t, for the (K  1)st degree B-spline functions. The knot vector t basically contains angles, h, between 0 and 2p in the form

ðpÞ

hl

where

ðbÞT

uk ¼ Vk qðbÞ ; ðA1Þ

ðA2Þ

ðbÞ

hl

ðbÞT

¼ Vhl qðbÞ

ðk ¼ x; y; z; l ¼ x; yÞ

ðB6a; bÞ

where ðbÞ

ðbÞ

ðbÞ

ðbÞ

ðbÞ

ðbÞ

ðbÞ

ðbÞ

ðbÞ

ðbÞ

ðbÞ

ðbÞ

qT ¼ fuxð0Þ ; uyð0Þ ; uzð0Þ ; hxð0Þ ; hyð0Þ ; uxðLÞ ; uyðLÞ ; uzðLÞ ; hxðLÞ ; hyðLÞ ; hxðmÞ ; hyðmÞ g ðB7Þ The explicit expressions for the vectors of known coefficient funcðbÞ ðbÞ tions, Vk ðk ¼ x; y; zÞ and Vhl ðl ¼ x; yÞ, are given in Appendix C.

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A. Barut, E. Madenci / Composite Structures 88 (2009) 579–594

Furthermore, the vector representation of the contact stress distribution in the circumferential direction along the contact region is given in the form ðpÞ

k ðuÞ ¼ k

ðb;pÞT

ðpÞ

K ðuÞ

ðB8Þ

where

¼ fk1 ; k2 ; . . . ; kL g; n o ðpÞ ðpÞ K ðuÞ ¼ KðpÞ 1 ðuÞ; K2 ðuÞ; . . . ; KLc ðuÞ k

ðpÞ

T

BðpÞ CðpÞ BðpÞ dA

A

Z Z T T T 1 ðbÞT ðbÞ BðbÞ CðbÞ BðbÞ dzqðbÞ þ qðbÞ BðbÞ  s0 dz q 2 ‘ ‘ T T 1 ðbÞ ¼ qðbÞ KðbÞ qðbÞ þ qðbÞ  Pcl 2

ðB10Þ

Using the matrix representations of the displacements and slopes in the laminates and the bolt, the strain resultants in the laminates and bolt and the shearing and peeling strains in the adhesive are expressed in matrix form as

e

ðpÞ

ðpÞ

¼B q ;

e

ðbÞ

ðbÞ

KðbÞ ¼

Z

T

BðbÞ CðbÞ BðbÞ dA



ðbÞ

Pcl ¼

¼B q ; ðp ¼ 1; 2Þ

ðB11a—cÞ

where explicit expressions for the transformation matrices BðpÞ ; BðbÞ , and Bða;lÞ ðl ¼ 1; 2Þ are given in Appendix C. Substituting into Eqs. (7), (19), and (12) from Eq. (B.11a–c), respectively, the resultant stresses in the laminates, the resultant forces in the bolt, and the shear and peel stress components in the adhesive are expressed in terms of the unknown generalized coordinates as

ðB19Þ

ðpÞ

and the internal forces arising from pre-stressing (tensioning) of the ðbÞ bolt by s0 are defined in the form

ðbÞ

eðpÞ ¼ Bða;1Þ qð1Þ þ Bða;2Þ qð2Þ

Z

T

ðbÞ

BðbÞ  s0 dz

ðB20Þ

‘

Substituting the matrix representations of the adhesive strain vector, eðaÞ , from Eq. (B11c) into Eq. (14), the strain energy of the adhesive becomes

Z 2 X 2 X T 1 ðpÞT Bða;pÞ EðaÞ Bða;lÞ dAqðlÞ q ðpÞ 2 A p¼1 l¼1 ( )" ð1;1Þ T 2 X 2 X Ka 1 ðpÞT ðp;lÞ ðlÞ 1 qð1Þ ¼ q Ka q ¼ T ð2ÞT 2 2 q Kð1;2Þ p¼1 l¼1

U ðaÞ ¼

a

ðbÞ

ðbÞ

sðbÞ ¼ CðbÞ eðbÞ þ s0 ¼ CðbÞ BðbÞ qðbÞ þ s0

ðB12a—cÞ

rðaÞ ¼ EðaÞ e ¼ EðaÞ ðBða;1Þ qð1Þ þ Bða;2Þ qð2Þ Þ

h i T T ðpÞT ðbÞ DU ðb;pÞ ¼ VðbÞ  VðpÞ  ðz  zðpÞ ÞVhn qðpÞ n q n c  2q0 ð1  cosðu  u0 ÞÞ ¼ 0

ðB13Þ

on u1 6 u 6 u2 . Similarly, evaluating the displacement and slope components of both the laminates and bolt in the bolt head–laminate contact regions at the bottom and top surfaces of the joint and rearranging the resulting matrix equations yield T

ðbÞT

T

VL qðbÞ  Vð2Þ qð2Þ ¼ 0 z

ðB14a; bÞ

¼

þ ðx  x

ðbÞ

ðbÞ

ðbÞ ÞVhx ð0Þ

þ ðy  y

ðbÞ

ðbÞ

ðbÞ ÞVhy ð0Þ ðbÞ

ðbÞ ðbÞ VL ¼ VðbÞ z ðLÞ þ ðx  x ÞVhx ðLÞ þ ðy  y ÞV hy ðLÞ

U ðpÞ ¼

1 ðpÞT q 2 ðpÞ

Z AðpÞ

T

BðpÞ CðpÞ BðpÞ dAqðpÞ ¼

1 ðpÞT ðpÞ ðpÞ q K q 2

1 ¼ 2

X

)

qð2Þ

ðB22Þ

ðaÞ

( (

qðbÞ

T

qð1Þ

T

qðbÞ

T

qð1Þ

T

)"

ðb;bÞ

K0

ðb;1ÞT

)"

K0

ðb;bÞ

KL

ðb;2ÞT

KL

ðb;1Þ

K0

#(

qð1Þ

ð1;1Þ

K0

ðb;2Þ

KL

qðbÞ

#(

ð2;2Þ

KL

qðbÞ qð1Þ

) ; ) ðB23a; bÞ

in which the submatrices of stiffness coefficients of the adhesive bond are defined as

ðB15bÞ

KL

ðb;bÞ

ðb;1Þ

K0

ð1;1Þ

K0

where K represents the matrix of the stiffness coefficient of the laminates and is defined in the form

A

ðb;2Þ cl

X

ðB15aÞ

ðB16Þ

T

Bða;pÞ EðaÞ Bða;lÞ dA

1 ¼ 2

K0

Substituting for the matrix representation of the strain resultants of laminates from Eq. (B11a) into Eq. (9), the strain energies of the laminates are expressed in matrix form in terms of their unknown generalized coordinates as

Z

ðb;1Þ cl

ðb;bÞ

VðbÞ z ð0Þ

qð1Þ

Note that Kð1;1Þ and Kð2;2Þ represent the self-stiffness coefficient a a matrices of the adhesive, which provide additional stiffness to each provides the coupling between individual laminate, whereas Kð1;2Þ a the two laminates in the lap joint. Strain energies of the springs placed between the bolt head– laminate contact area are obtained by substituting from Eq. (B14) into the strain energy expressions in Eq. (28) as

where ðbÞ V0

Kað2;2Þ

#(

with

Kðp;lÞ ¼ a

Substituting for the displacement and slope components of the laminate and bolt along their contact region around the laminate hole boundary, the constraint equation coupling the bolt and the laminate displacements and slopes, Eq. (24), is rewritten in matrix form, after rearranging the terms, as

ðbÞT

Kað1;2Þ

ðB21Þ

sðpÞ ¼ CðpÞ eðpÞ ¼ CðpÞ BðpÞ qðpÞ

V0 qðbÞ  Vð1Þ qð1Þ ¼ 0; z

ðB18Þ

where KðbÞ is the matrix of beam (bolt) stiffness coefficients defined as

!

A

ðpÞ

ðB17Þ

ðpÞ

Similarly, substituting for the matrix representation of the strain resultants of the bolt from Eq. (B11b) into Eq. (21), the strain energy of the bolt is expressed in matrix form in terms of its unknown generalized vector, qðbÞ , as

ðB9a; bÞ

with

u  uðpÞ 1 ðpÞ uðpÞ 2  u1

Z

U ðbÞ ¼

ðb;pÞT

KðpÞ n ðuÞ ¼ sin np

KðpÞ ¼

¼

Z Acl

¼

Z

Acl

ðbÞ

ðbÞT

ðbÞ

ðbÞT

V0 V0 VL VL

dA; dA;

ðb;1Þ

K0

¼

Z T ðbÞ ¼ V0 Vð1Þ dA; z Acl Z Z ð2;2Þ ð1ÞT ¼ Vð1Þ V dA; K ¼ z z L Acl

Z

T

ðbÞ

Acl

V0 Vð1Þ dA z

T

Acl

ð2Þ Vð2Þ dA z Vz

ðB24a—cÞ

ðB24d—fÞ

where Acl denotes the clamp-up area under the bolt heads. Similar to the adhesive strain energy expression in Eq. (B21), the stiffness ðb;bÞ ð1;1Þ ðb;bÞ ð2;2Þ yield self-stiffness coeffisubmatrices K0 ; K0 ; KL , and KL

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A. Barut, E. Madenci / Composite Structures 88 (2009) 579–594

cient matrices that are added to the self-stiffness matrices of the ðb;1Þ ðb;2Þ bolt and the laminates, whereas K0 and K0 represent the coupling stiffness coefficient submatrices between the bolt and the laminates. Since both laminates are coupled with the same bolt through the bolt head–laminate contact conditions, the laminates are also coupled in an indirect way. The strain energy of the elastic supports under prescribed displacements and slopes around the laminate edges, ‘ðpÞ , is obtained, after substituting from Eq. (B1) for the laminate displacements, as 2 2 X T T 1X ðpÞ ¼ qðpÞ KX qðpÞ  qðpÞ  PðpÞ u 2 p¼1 p¼1

XðpÞ

ðB25Þ

Gðb;pÞ ¼

X Z

¼

ðpÞ

‘

j¼x;y;z  ðpÞ Pu

ðpÞ ðpÞT kj Vj Vj

j¼x;y

X Z

¼

ðpÞ ðpÞ kj Vj  uj d‘

‘ðpÞ

j¼x;y;z

d‘ þ

XZ

þ

‘

ðpÞ ðpÞT khj Vhj Vhj

ðpÞ

XZ

j¼x;y

‘ðpÞ

ðpÞ

khj Vhj  hj d‘

Similarly, the strain energy of the springs attached between the laminates and the rigid load fixture [Fig. 4 (bottom)] in Eq. (31) is rewritten in matrix form as



2   T T 1X ðpÞ T ðgp Þ qðpÞ KðpÞ gp  qðpÞ Kðp;gp Þ gp  gTp Kðgp ;pÞ qðpÞ g q þ gp K 2 p¼1

XZ

ðpÞT

ðpÞ

‘ðpÞ

kgj Vj Vj

Kðgp ;gp Þ ¼

d‘;



Z ‘ðpÞ

kgx

0

0

kgy



KT ½1  cosðu  u0 Þdu

ðB33cÞ

2 X

qðpÞ

T

Z

T

AðpÞ

 VðpÞ p0 dA ¼ qðpÞ  PðpÞ z pr

ðB33dÞ

PðpÞ pr ¼

Z

 VðpÞ p0 dA z

AðpÞ

ðB33eÞ

Also, the line loads along the laminate edges, ‘ðpÞ , in Eq. (34) are rewritten in the form Vr ¼

2 X

T

qðpÞ

Z ‘

p¼1 2 X

 ðpÞ

 ðpÞ ðpÞ  ðpÞ ðpÞ  ðpÞ  ðpÞ  ðpÞ VðpÞ Nx þ VðpÞ Nz þ Vhx  mðpÞ d‘ x þ V hy my x y N y þ Vz

T

qðpÞ  PðpÞ r

where

d‘ ðB28a; bÞ

Z

Kðp;gp Þ ¼

h ðpÞ

kgx VðpÞ x

i

d‘; kgy VðpÞ y

Kðgp ;pÞ ¼ Kðp;gp Þ

T

ðpÞ

zðpÞ h

Z

u2

u1

 h i  T T ðpÞT ðbÞ ðpÞ qðpÞ  qn rdu dz kðpÞ VðbÞ  VðpÞ n q n ðzz ÞV hn

As mentioned before, the contact stresses, kðpÞ , are assumed to be uniform through the laminate thicknesses, i.e., kðpÞ ¼ kðpÞ ðhÞ. Eq. (B29) then reduces to u2

u1

  T T ðpÞ kðuÞ Vðb;pÞ qðbÞ  VðpÞ  qn r du  0 n n q

ðB30Þ

where

Z

Vðb;pÞ ¼ n

zðpÞ þhðpÞ

zðpÞ hðpÞ

VðbÞ n dz

ðB31Þ

Substituting into Eq. (B30) from Eq. (B8) for the vector representation of the contact stresses and rearranging the terms, the potential energy of the constraint forces (resultant contact stresses) are expressed in compact form as ðb;pÞT

¼k

ðb;pÞ

G

ðbÞ

q

ðb;pÞT

þk

ðpÞ

G q

ðpÞ

ðb;pÞT

k

ðb;pÞ

Pq

0

Z ‘

  ðpÞ ðpÞ  ðpÞ ðpÞ  ðpÞ  ðpÞ  ðpÞ d‘ VðpÞ Nx þ VðpÞ Nz þ Vhx  mðpÞ x þ V hy my x y N y þ Vz

ðpÞ

Furthermore, in the case of concentrated loads acting on the rigid load fixture [Fig. 4 (bottom)], the potential energy expression defined in Eq. (32) is rewritten in matrix form as

Vp ¼

ðB32Þ

2 X

T

DðpÞ  PðDp Þ

ðB33hÞ

p¼1

where 

ðB29Þ

Z

PðpÞ r ¼

ðB33gÞ

Substituting the vector representation of the displacements and slopes of the laminates and the bolt from Eqs. (B1) and (B6), the potential energy of the constraint forces (contact stresses) acting on the surface of the contact region, as given in Eq. (25), is expressed in vector notation as zðpÞ þhðpÞ



ðB28c; dÞ

‘

where

u1

p¼1

j¼x;y

V ðpÞ c

u2

where

¼

Kðp;pÞ ¼ g

V ðpÞ c ¼

Z

ðB33a; bÞ

Using Eq. (C1) for the laminate displacement field, the potential enðpÞ ergy of the pressure loading, p0 , acting on the surface of the laminates in the vertical (z-) direction in Eq. (33) is rewritten in matrix form as

where

Z

T

ðbÞ KT VðpÞ du; n r

Pðb;pÞ ¼ 2q0 rðbÞ q

ðB27Þ

V¼

u2

u1



XðpÞ g ¼

T

KT Vðb;pÞ r ðbÞ du; n

p¼1

ðB26a; bÞ ðpÞ

Z

GðpÞ ¼ 

V pr ¼

d‘

u2 u1

where

KðpÞ u

Z

T

Pð D p Þ ¼

n

 ðDp Þ  ðDp Þ Px ; Py

o

ðB33iÞ

Appendix C In Eq. (B1), the vectors containing the known functions of the assumed solution forms for the local displacement and slope fields ðpÞ ðj ¼ x; yÞ, as defined in Eq.  ðpÞ of laminates, u k ðk ¼ x; y; zÞ and hj (37), are expressed in the form

n o T T VðpÞ ¼ VðpÞ ; 0T ; 0T ; 0T ; 0T ; x e n o T T T T VðpÞ ¼ 0T ; 0T ; VðpÞ z j ;0 ;0 n o T T ðpÞT Vhx ¼ 0T ; 0T ; 0T ; VðpÞ 0 ; e

n o T T VðpÞ ¼ 0T ; VðpÞ ; 0T ; 0T ; 0T ; y e ðC1a—cÞ n o T ðpÞT Vhy ¼ 0T ; 0T ; 0T ; 0T ; VðpÞ e ðC1d; eÞ

where

n o T VðpÞ ¼ ln rTT ; rNr TT ; r Nr þ1 TT ; . . . ; r 1 TT ; rTT ; . . . ; r Nr 1 TT ; r Nr TT e ðC2aÞ

594

A. Barut, E. Madenci / Composite Structures 88 (2009) 579–594

T r Nr þ1 T r Nr þ2 T r2 T VðpÞ ¼ rðln r  1ÞTT ; ln rST ; T ; T ;...; T ; j N r þ 1 N r þ 2 2

r2 rNr T r Nr þ1 T rTT ; TT ; . . . ; ðC2bÞ T ; T 2 Nr Nr þ 1 where the subvector T ½¼ TðhÞ] contains the coefficients of the periodical B-spline functions, T i , defined in the circumferential direction of the polar coordinate system located at the center of the cutout; they are described in detail in Appendix A. The coefficient vectors associated with the global representa ðpÞT and V  ðpÞT , which contain the tion of the displacement fields, V hj k Bernstein polynomial terms and appear in Eq. (B2), are expressed as

n o T T T T ; ¼ VðpÞ e ;0 ;0 ;0 ;0 n o T T T T ðpÞ ¼ 0 ; 0 ; Ve ; 0 ; 0

VðpÞ x VðpÞ z

n o ðpÞ T ; Vhx ¼ 0T ; 0T ; 0T ; VðpÞ e ;0

VðpÞ y

n o T T T ; ¼ 0T ; VðpÞ e ;0 ;0 ;0

n o ðpÞ Vhx ¼ 0T ; 0T ; 0T ; 0T ; VðpÞ e

In Eq. (B11), the transformation matrices between (resultant) strains and displacements/slopes in the laminates and the bolt contain

2

2 6 6 BðpÞ e ¼ 6 4

ðC3a—cÞ

4 BðpÞ c ¼

ðC3d; eÞ

and

ðC4bÞ

in which the Bernstein polynomials are already defined by Eq. (40). ðbÞ ðbÞ In Eq. (B6), the vectors Vk ðk ¼ x; y; zÞ and Vhl containing the known functions of the assumed displacement and slope fields of ðbÞ  ðbÞ ðk ¼ x; y; zÞ and  hj ðj ¼ x; yÞ, as given in Eq. (41), are the bolt, u k defined as T

z 5z 3z2 2z3 z z z2 2z3 1  ; 0; 0;  þ 2 ; 0; ; 0; 0;   þ 2 ; 0; L 6 L 6 2L 3L 2L 3L

2z 2z2 4z3  þ ðC5aÞ  2 ;0 3 L 3L

VðbÞ ¼ x

T

z 5z 3z2 2z3 z z z2 2z3 0; 1  ; 0; 0;  ; 0; ; 0; 0;   þ 2 ; 0; þ L 6 L 6 2L 3L 2L 3L2

2z 2z2 4z3  þ  2 ðC5bÞ 3 L 3L

VðbÞ ¼ y

n o T z z ¼ 0; 0; 1  ; 0; 0; 0; 0; ; 0; 0 VðbÞ z L L ðbÞT

Vhx ¼

ðbÞT

Vhy ¼

ðC5cÞ



2z2 3z 2z2 z 4z2 4z 0; 0; 0; 2  þ 1; 0; 0; 0; 0; 2  0;  2 þ ; 0 L L L L L L ðC5dÞ

2z2 3z 2z2 z 4z2 4z 0; 0; 0; 0; 2  þ 1; 0; 0; 0; 0; 2  ; 0;  2 þ L L L L L L ðC5eÞ

ðC6Þ

where

2

ðC4aÞ

3

6 ðpÞ 7 7 BðpÞ ¼ 6 4 Bj 5 ðpÞ Bc

VðpÞ x;x

T

VðpÞ y;y

T

3

2

7 7 7; 5

6 6 BðpÞ j ¼6 4

T

T

T

ðpÞ

3 T

T

ðpÞT

ðpÞ VðpÞ x;y þ Vy;x

VðpÞ z;x þ Vhx

VðpÞ z;y þ Vhy

3

ðpÞT

Vhx;x

ðpÞT

Vhy;y ðpÞT

ðpÞT

7 7 7; 5

Vhx;y þ Vhy;x

5

ðC7a — cÞ

3 VðbÞ z;z 7 6 7 6 VðbÞ hx;z 7 6 7 6 ðbÞ 7 6 ¼ 6 Vhy;z 7 7 6 ðbÞ 6 V þ VðbÞ 7 hx 5 4 x;z 2

where

n o ðpÞ ðpÞ ðpÞT ðpÞ ðpÞT ðpÞT VðpÞ e ¼ B0 ðt x ÞBy ; B1 By ; . . . ; BMx ðt x ÞBy n o T ðpÞ ðpÞ ðpÞ BðpÞ ¼ B0 ðt y Þ; B1 ðt y Þ; . . . ; BMy ðty Þ y

BðpÞ e

BðbÞ

ðC8Þ

ðbÞ

VðbÞ x;z þ V hy

In the adhesive, the transformation matrices Bða;pÞ ðp ¼ 1; 2Þ in Eq. (B12) are defined as

2 Bða;1Þ ¼

Vð1Þ z

3

7 1 6 6 Vð1Þ þ hð1Þ Vð1Þ 7; hx 5 x ðaÞ 4 ð1Þ ð1Þ Vð1Þ y þ h Vhy

2h

2 Bða;2Þ ¼

Vð2Þ z

3

7 1 6 6 Vð2Þ þ hð2Þ Vð2Þ 7 hx 5 ðaÞ 4 x ð2Þ ð2Þ Vð2Þ y þ h V hy

2h

ðC9a; bÞ References [1] Hart-Smith LJ. Bolted–bonded composite joints. J Aircraft 1985;22(11):993–1000. [2] Mathews FL. Joining fibre-reinforced plastics. New York, NY: Elsevier Applied Science; 1987. [3] Sarkani S, George M, Kihl DP, Beach JE. Stochastic fatigue damage accumulation of FRP laminates and joints. J Struct Eng 1999;125(12):1423–31. [4] Fu M, Mallick PK. Fatigue of hybrid (adhesive/bolted) joints in SRIM composites. Int J Adhes Adhes 2001;21(2):145–59. [5] Lees JM, Makarov G. Mechanical/bonded joints for advanced composite structures. In: Proceedings of the institution of civil engineers – structures and buildings, vol. 157; 2004. p. 91–7. [6] Lin W-H, Lin C-H, Jen M-HR. An innovative fatigue method in single-lap mixed joints. J Reinforced Plastic Compos 2004;23(9):997–1017. [7] Kradinov V, Barut A, Madenci E, Ambur AR. Bolted double-lap composite joints under mechanical and thermal loading. Int J Solid Struct 2001;38(44– 45):7801–37. [8] Oterkus E, Barut A, Madenci E, Smeltzer III SS, Ambur DR. Bonded lap joints of composite laminates with tapered edges. Int J Solid Struct 2006;43(6):1459–89.