Open access peer-reviewed chapter

A Unified Analysis of Adhesive-Bonded Cylindrical Coupler Joints

By Sontipee Aimmanee

Submitted: July 13th 2017Reviewed: November 7th 2017Published: December 20th 2017

DOI: 10.5772/intechopen.72288

Downloaded: 421

Abstract

In the past years, many studies have been conducted on behaviors of adhesive tubular joints subjected to various loading conditions, such as torsion, axial, and internal and external pressure. However, the previous models are conceptually distinct, since they were developed to analyze only for each type of load. Mostly, homogeneous isotropic or orthotropic material were considered and thin-walled joint structures were examined. Therefore, the aim of this chapter is to present for the first time a generalized mathematical formulation and modeling of adhesive-bonded cylindrical coupler joints taking into account all loading scenarios. The inner and outer adherends can be made of isotropic, orthotropic, or laminated composite materials, and they are modeled as three-dimensional elastic body, so adherends with any thickness can be analyzed. Assumptions of an axisymmetric joint with linearly elastic adherends and adhesive materials are employed. Thin adhesive layer is considered so that only the out-of-plane adhesive stresses are concerned, and they are treated to be uniform through its thickness. Using elasticity theory and the newly developed finite-segmented method, stress distributions in both adherends and adhesive can be evaluated. Calculation examples of laminated composite joints are given. This model provides the unified analysis of adhesive-bonded cylindrical coupler joints.

Keywords

  • adhesive
  • coupler joint
  • lap joint
  • elasticity
  • finite-segment method

1. Introduction

Structures usually need to have joints connecting each part together due to the limitation of manufacturing, transportation, and installation. These structures are generally vulnerable at the joints because of the stress concentrations from material discontinuity. There are many types of joint, such as mechanical joints, welding joints, and adhesive-bonded joints. Over the other kinds of joints, adhesive-bonded joints have advantages due to less stress concentration, higher capability of joining dissimilar materials, lighter weight, and better corrosion resistance. Nonetheless, stress concentration in the adhesive layer is still existing. The adhesive stress distribution is locally nonuniform and always highest at the edges of the bonding region. Thus, in order to use an adhesive-bonded joint safely, it is important to predict the developed adhesive stress accurately. A good analysis also provides the understanding of the joint behavior, yielding a design for improving the joint performance by decreasing the joint stress concentration.

There are lots of literature dealing with the stresses in lap joints and coupler joints between two tubular adherends. Among several types of adhesive-bonded joint, cylindrical or tubular joints subjected to axial, torsional, and external as well as internal pressure loads have been one of the main focus of mechanics of adhesion research for a half century due to their popular usage in many engineering applications. The analytical modeling and finite-element analysis are the two popular approaches for predicting the stresses developed in adhesive and adherends. In order to recognize the progress in this field, some examples of important work in mathematical modeling are given below.

For axial loads, the early investigation was conducted by Lubkin and Reissner [1] who analyzed the tubular lap joints. Axial shear stresses and radial normal stresses in the adhesive layer were predicted under the assumption of approximating the layer as an infinite number of tensile and shear springs. However, disappearance of axial shear stress on the free surfaces at the ends of the adhesive was not considered. Adam and Peppiatt [2] performed axisymmetric finite-element analysis (FEA) of tubular lap joints subjected to stretching and twisting loads. The effects of an adhesive filet and partial tapering adherends on stress distribution were also reported. Using a minimum strain energy, Allman [3] proposed two-dimensional analytical solution for lap joints that ensure the traction-free boundary condition. Bending, stretching, and shearing of the adherends and shearing and tearing of the adhesive layer were taken into account. Shi and Cheng [4] formulated closed-form solutions for tubular lap joints utilizing the variational principle of complementary energy. Boundary conditions and assumptions of Allman were adopted in their model development. Nemes et al. [5] further developed the stress analysis of adhesive in a cylindrical assembly of two tubes. Variational method of the potential energy was also employed. Nonetheless, Nemes et al. neglected radial stress component in the joint. Kumar [6] presented a theoretical framework for the stress analysis of shaft-tube adhesive joints subjected to tensile loads. The joint assembly was considered to consist of similar or dissimilar isotropic or orthotropic adherends. The principle of minimum complementary energy and a stress function approach were used to establish the governing equations in order to determine the stress state in each constituent. To reduce the stress concentration, Kumar [7] also studied the use of functionally graded adhesive in a tubular lap joint with an isotropic adherend under tension. In his model, the adhesive was divided into annular rings to take into account the gradient property of shear modulus.

Regarding the case of the tubular joint subjected to torsion, Volkersen [8] provided a closed-form solution for circumferential shear stresses at the interface of tubular lap joint exerted by a torque. Pugno and Surace [9] investigated the analysis of the joint subjected to torsion. They utilized the common function of resultant torques in adherends and achieved the uniform adhesive hoop shear stress by tapering adherend surfaces. Xu and Li [10] investigated the full three-dimensional stress analysis of a bonded tubular-coupler joint subjected to torsion. Their purpose was to investigate all of the adherend and adhesive stress components without the assumption of through-thickness constant stresses across the adhesive layer. Oh [11] performed an analysis of the bonded tubular lap joint of laminated tubes with softening adhesive’s stiffness properties under torsion using an elasticity model. Oh concluded that the load capacity in the linear analysis can be quite underestimated when compared to the nonlinear modeling. Spaggiari and Dragoni [12] investigated the joint studied in the Kumar’s work in [7], but the joint is subjected to torsion instead. They developed the closed-form function of the adhesive shear modulus in order to minimize adhesive shear stress over the bonding region and addressed the limitation of shear modulus and thicknesses ratio for joint manufacturing with functionally graded adhesives. Recently, Aimmanee and Hongpimolmas [13] formulated a mathematical model of an adhesive-bonded tubular joint with a variable-stiffness composite coupler. The optimal variable fiber orientation in the coupler was determined to minimize the adhesive hoop shear stress.

Stresses in cylindrical joints under pressure have also been studied, even though investigation of this type of load was comparatively scarce compared to the above two loadings. Terekhova and Skoryi [14] provided a close-form solution for the stresses in tubular lap joints under external and internal pressures and axial forces. Their model neglected the effect of adherend bending. Baishya et al. [15] conducted research in individual and combined effect of internal pressure and torsional loading on stress and failure characteristics of tubular single lap joints made of composite materials. The onset of different joint fracture modes was investigated in their work. Strength analysis of adhesive joints of riser pipes in deep sea environment loadings was performed by Zhang et al. [16] External pressure, internal pressure, tension, torsion, and bending were examined to understand singular stress fields existing around end of the interface. Apalak [17] investigated elastic stresses in the adhesive layer and tubes of an adhesively bonded tubular joint with functionally graded tubes subjected to an internal pressure. Finite-element method was used to model the tubes having gradient layer between a ceramic layer and a metal layer.

According to the former analytical research work presented in the literature, the problems can be mathematically complicated even though the joint is made of simple conventional isotropic adherends. In addition, the previous models usually are distinct for each type of load, since they were developed to analyze only for a specific loading case. Therefore, this chapter aims to present a mathematical modeling of adhesive-bonded cylindrical coupler joints taking into account all loading scenarios, i.e., torsion, axial, and pressure loadings. The inner and outer adherends can be made of isotropic, orthotropic, or laminated composite materials, and they are modeled as three-dimensional elastic body, so a thick or solid cylinder adherend can also be analyzed. Stresses in adhesive layer and adherends can be evaluated by newly developed finite-segment method. The unified formulation of the model will be discussed in the next section.

2. Elasticity theory of a laminated cylindrical structure

A bonded-coupler joint is illustrated in Figure 1(a). The joint consists of two inner tubes (adherend part 1), a coupler of length 2 L (or adherend part 2), and an adhesive layer. The cylindrical coordinates xθrdepicted in Figure 2 are used to describe the joint geometry. Because of symmetry about the cross-sectional plane in the middle, only half of the coupler joint is demonstrated in Figure 1(b). Either half is equivalent to a single tubular lap joint and applicable for modeling and analysis. The inner tubes are considered to be made of an ordinary material, such as an isotropic metal or a more sophisticated material, namely, orthotropic material or laminated composite. On the contrary the coupler is proposed to be fabricated from a symmetric-balanced laminated composite with variable fiber orientation in the x direction.

Figure 1.

Schematic of a bonded-coupler joint: (a) full model and (b) half model or tubular lap joint model.

Figure 2.

A laminated tube and the defined coordinate systems.

For the sake of generality, this section discusses the elasticity theory of a laminated cylindrical tube [18]. A sketch of a general open-ended, cylindrical, laminated N-layer tube subjected to uniform loads is shown in Figure 2. Each layer is made of a unidirectional fiber-reinforced composite material. The principal material coordinates (1, 2, 3), whose axes are mutually orthogonal, are defined along the fiber orientation, tangent, and normal to the tube surface, respectively. The layers in the tube are perfectly bonded between each other. Evidently, this considered laminated cylinder can be simply degenerated into a single isotropic or orthotropic tube by letting N = 1 and employing the related elastic properties. For the tube with axisymmetric geometry and circumferentially independent material properties under a uniform load, the strain–displacement relations in the kth layer in the cylindrical coordinates are

εxk=ukx,εθk=wkr,εrk=wkrγθrk=vkrvkr,γxrk=ukr,γk=vkxE1

where εand γdenote normal and shear strains, respectively. u, v, and w are displacements in axial, tangential, and radial directions, respectively. Superscript (k) indicates that the corresponding quantities are in the kth layer.

According to the prescribed loading conditions and constant fiber orientation, the normal stresses, σ, and the shear stresses, τ, are independent of x and θ. The equilibrium equations in the kth layer along the r-, θ-, x-directions are reduced to ordinary differential equations with respect to r, respectively, as

σrkr+1rσrkσθk=0E2
τθrkr+2rτθrk=0E3
τxrkr+1rτxrk=0E4

The stresses and strains in the kth layer in xθrcoordinates expressed in Eqs. (1)(4) can be transformed to those in the principal material coordinates (1, 2, 3) as follows:

σ123k=Tkσxθrk,ε123k=TkεxθrkE5

where σ123kand ε123kare tensorial stress and tensorial strain components, respectively. Tkis transformation matrix of the kth layer as shown in Eq. (6), in which mk=cosØkand nk=sinØk. Økis fiber angle of the kth layer as shown in Figure 2.

Tk=mk2nk20002mknknk2mk20002mknk001000000mknk0000nkmk0mknkmknk000mk2nk2E6

The constitutive relation in the kth layer in the cylindrical coordinates can be written as

σxθrk=C¯kεengxθrkE7

In the above, C¯kis the transformed stiffness matrix, and εengxθrkis engineering strain components in the global cylindrical coordinate system. The transformed stiffness matrix C¯kcan be evaluated as

C¯k=Tk1CkRTkR1E8

where Ck, as shown in Eq. (9), is the stiffness matrix in the principle material coordinate system in the kth layer.

Ck=1/E1kυ12k/E1kυ31k/E3k000υ21k/E2k1/E2kυ32k/E3k000υ13k/E1kυ23k/E2k1/E3k0000001/G23k0000001/G13k0000001/G12k1E9

Ekand Gkare Young’s modulus and shear modulus, respectively. Ris the Reuter’s matrix, which is defined as

R=100000010000001000000200000020000002E10

With the strains defined in Eq. (1), three out of six equations of the compatibility in the cylindrical coordinates described in [19] are automatically satisfied. Solving the equilibrium equations in Eqs. (2)(4) and using the strain–displacement relations in Eq. (1), the constitutive relation in Eq. (7), the remaining three compatibility equations, as well as the displacement continuity between each layer yield the displacement expressions in the kth layer of the laminated tube as illustrated in Eqs. (11)(13):

ukx=εx0xE11
vkxr=γ0xrE12
wkr=A1krλk+A2krλk+Γkεx0r+Ωkγ0r2E13

In the above, εx0and γ0are axial strain and angle of twist per unit length constants, respectively. A1kand A2kare the integration constants in the kth layer. λk, Γk,and Ωkare described in Eq. (14) in terms of components of the transformed stiffness matrix in the kth layer:

λk=C¯22kC¯33k,Γk=C¯12kC¯13kC¯33kC¯22k,Ωk=C¯26k2C¯36k4C¯33kC¯22kE14

Note that when a layer is made of 0o fiber orientation, the stiffness coefficients C¯16k, C¯26k, and C¯36kare zero and C¯12k=C¯13kas well as C¯22k=C¯33k. As such, Eq. (13) is degenerated to become wkr=A1krk+A2kr1k[13].

For an N-layer laminated tube, there are 2 N + 2 unknown integration constants to be evaluated. Therefore, 2 N + 2 equations are required to solve for the constants. The first four equations written in Eqs. (15)(18) are obtained from two equations of the force equilibrium with the external loads and two equations from the surface traction boundary conditions. Note that F in Eq. (15) is an axial force, T in Eq. (16) an applied torque, piin Eq. (17) a normal traction or internal pressure on the inner surface, and poin Eq. (18) a normal traction or external pressure on the outer surface. The remaining 2 N-2 equations can be obtained from N-1 continuity conditions of the interfacial radial stresses, σr, and N-1 continuity conditions of the interfacial radial displacements, w, as shown in Eqs. (19) and (20), respectively:

R0RN2πσxrdr=2πk=1NRk1Rkσxkrdr=FE15
R0RN2πτr2dr=2πk=1NRk1Rkτkr2dr=TE16
σr1R0=piE17
σrNRN=p0E18
σrkRk=σrk+1Rk(k=1,2,3,…,N-1)E19
wkRk=wk+1Rk(k=1,2,3,…,N-1)E20

Eqs. (15)(20) give the system of algebraic equations written in matrix form as

k11k12k13k21k22k23k31k41k32k42k33k43k1,2N+2k2,2N+2k3,2N+2k4,2N+2k2N+2,1k2N+2,2k2N+2,3k2N+2,2N+2FTpipo000=ϵx0γ0A11A21A1NA2NE21

where kijij=1232N+2are the coefficients obtained from the equations above. By solving Eq. (21), the constants ϵx0,γ0,and A11,A21,,A1N,A2Ncan be obtained. Subsequently, all displacements, strains, and stresses are calculated by using Eqs. (11)(13), (1), and (7), respectively.

3. Formulation of an equivalent lap joint model

3.1. Derivation of governing equations

All geometric parameters of a perfectly bonded tubular lap joint are shown in Figure 1(b). The adhesive is assumed to be isotropic and linearly elastic. The adhesive thickness tais considered to be very thin compared to the adherend thicknesses, and thus, the outer radius of part 1 R1ois approximately the same as the inner radius of part 2, R2i. In addition, there are only three out-of-plane stress components mainly contributed in the adhesive: hoop shear stress τθra,longitudinal shear stress τxra, and radial normal stress σra. These stresses in the adhesive are treated to be uniform through the adhesive thickness. Applied torque T; applied axial force F; internal pressure exerted on the inner surface of adherend part 1, pior written specifically as p1i; and external pressure exerted on the outer surface of adherend part 2, poor p2o,are all included in the following formulation.

In order to derive the governing equations, let us initially consider the torque transmission through a coupler joint. The applied torque T is assumed to distribute only in the adherend part 1 and adherend part 2 as denoted as T1and T2, respectively. Hence, the applied torque T can be written as

T=T1+T2E22

To determine the variation of the T2along the bonding length, the adherend 2 is divided into elements with an infinitesimal length dx. The equilibrium between the resultant torque in the element and the adhesive hoop shear stress can be expressed as follows:

12πR2i2dT2xdx=τθra=GaγθraE23

In Eq. (23), Gais shear modulus of adhesive. By considering the deformation of an adhesive element on a cross-sectional plane in the overlap region of the perfectly bonded joint, the kinematic condition in the adhesive can be written as

γθra=v2iv1otaE24

and its derivative with respect to x is.

dγθradx=γ2iγ1otaE25

Combining Eqs. (23) and (25) yields the first governing equation:

d2T2xdx2=2πR2i2Gataγ2iγ1oE26

Next, consider equilibrium of resultant axial force. When the joint is subjected to tension or compression loads, the resultant axial force in the adherend 1, F1,and in adherend 2, F2,are produced at any given cross section in the overlap region, similar to Eq. (22). The force equilibrium is

F=F1+F2E27

The variation of the F2along the length can be examined by considering an infinitesimal elements in adherend part 2 with the differential length dx. The equilibrium between resultant axial force in the element and the adhesive longitudinal shear stress τxracan consequently be expressed as follows:

12πR2idF2xdx=τxra=GaγxraE28

By considering compatibility of the joint, it can be shown that.

γxra=u2iu1otaordγxradx=εx2iεx1otaE29

Combining Eqs. (28) and (29) yields the axial force governing equation:

d2F2xdx2=2πR2iGataεx2iεx1oE30

Next, interacting through the adhesive thickness, the resultant normal traction acting on the outer surface of adherend 1, p1o, and that exerting on the inner surface of adherend 2, p2i,are generated. Under the assumption of thin adhesive layer, the resultant normal tractions p1oand p2iare related to each other as

p1o=p2iE31

Lastly, instead of directly equating adhesive radial normal stress to normal traction in (31), σracan be more accurately determined by the equilibrium equation in cylindrical coordinates of the adhesive layer showing in Eq. (32):

σrr+1rσrσθ+1rτθrθ+τxrx=0E32

With the conditions of axisymmetry, the equilibrium equation is reduced to

1R2iσraσθa+τxrax=0E33

According to the study conducted in [6], σθais observed to have the same distribution as σraso they are legitimately regarded as being proportional to each other via adhesive normal stress ratio α. Their relation can be mathematically expressed in Eq. (34):

σθa=ασraE34

As a consequence, the equilibrium equation in Eq. (32) can then be written as

σra=12π1αd2F2xdx2E35

3.2. Implementation of elasticity theory for adherends

The two governing equations Eqs. (26) and (30) have already been formulated to determine resultant loads in adherend part 2 of an adhesive-bonded-coupler joint. The resultant loads in adherend part 1 can be then calculated easily by using Eqs. (22) and (27) after all internal loads in adherend part 2 are evaluated. However, related through Eq. (21), the two equations are coupled and need to be solved altogether. To aptly deal with this complicated condition, the problem is separated into primary and secondary effects. When the joint is subjected to torsion, the hoop shear stress in the adhesive τθrais primary and dominant compared to the other adhesive stresses as discussed in [8, 10, 13], whereas in the case of the joint being under an application of longitudinal force, or external and internal pressure, the adhesive longitudinal shear stress τxraand adhesive radial normal stress σraare comparatively crucial [20]. By neglecting the secondary stress components and the corresponding resultant internal loads in the early calculation stage, the problem is then uncoupled and can be readily solved for the primary variables. The initially excluded stress components are later recovered by using the obtained solutions in the coupled set of governing equations.

First, further modification of the torque governing equation of Eq. (26) is performed, adherend in-plane shear strains γ1oand γ2imust be expanded in terms of the internal resultant loads. It can be seen that they are equal to γ01R1oand γ02R2i, respectively, where γ01and γ02are denoted for γ0of adherend parts 1 and 2. Utilizing Eq. (21) yields the relations:

R1ok211F1+k221T1+k231p1i+k241p1o=γ,F1oF1+γ,T1oT1+γ,pi1op1i+γ,po1op1o=γ1oE36
R2ik212F2+k222T2+k232p2i+k242p2o=γ,F2iF2+γ,T2iT2+γ,pi2ip2i+γ,po2ip2o=γ2iE37

where quantities k2X1and k2X2, where X=1,2,3, and 4 are the first four elements compliances in the second row of matrix kin Eq. (21). Superscripts 1 and 2 are defined for adherend parts 1 and 2, respectively. γ,F1o, γ,T1o, γ,pi1o, γ,po1o, γ,F2i, γ,T2i, γ,pi2i, and γ,po2iare in-plane shear strains per unit load on the outer interfacial surface of adherend part 1 and inner interfacial surface of adherend part 2, respectively.

By substituting Eqs. (36) and (37) into the governing equation Eq. (26) and utilizing Eqs. (22), (27), and (31), the governing equation in term of T2becomes

d2T2xdx2=KFF2x+KTT2x+Kpp2ix+KCE38

where the parameters KF, KT, Kp,and KCare.

KF=2πR2i2Gataγ,F2i+γ,F1o,KT=2πR2i2Gataγ,T2i+γ,T1o,Kp=2πR2i2Gataγ,pi2iγ,po1o,KC=2πR2i2Gataγ,po2ipoγ,pi1opiγ,F1oFγ,T1oTE39

Accompanying with the boundary conditions of Eq. (40), which are implied that torque in adherend part 2 is zero at x = 0 and fully transmitted at x = L, Eq. (38) is well-defined for solving resultant torque in the adherend part 2,T2:

T20=0,T2L=TE40

Second, analogous to Eqs. (36) and (37), adherend in-plane normal strains εx1oand εx2imust also be written in terms of the internal resultant loads. Again, using the expression in Eq. (21), the following expressions are obtained:

k111F1+k121T1+k131p1i+k141p1o=εx,F1oF1+εx,T1oT1+εx,pi1op1i+εx,po1op1o=εx1oE41
k112F2+k122T2+k132p2i+k142p2o=εx,F2iF2+εx,T2iT2+εx,pi2ip2i+εx,po2ip1o=εx2iE42

In the above, k1X1and k1X2, where X=1,2,3,and 4 are the first four elements in the first row of matrix kin Eq. (21). Superscripts 1 and 2 are defined for adherend parts 1 and 2, respectively. εx,F1o, εx,T1o, εx,pi1o, εx,po1oεx,F2i, εx,T2i, εx,pi2i, and εx,po2iare newly denoted to indicate the physical meaning of the parameters. They represent the in-plane normal strains due to unit load on the adhesive-interfacial surface in the adherends. The unit load quantities are distinguished by after-comma subscripts F, T, or p. Quantities εx1oand εx2iare εx0of adherends 1 and 2, respectively.

Combining the governing equations Eqs. (30), (41), and (42), as well as the load equilibriums in Eqs. (22), (27), and (31), yields a new form of the axial force governing equation:

d2F2xdx2=kFF2x+kTT2x+kpp2ix+kCE43

where the parameters kF, kT, kp, and kCare.

kF=2πR2iGataεx,F2i+εx,F1o,kT=2πR2iGataεx,T2i+εx,T1okp=2πR2iGataεx,pi2iεx,po1o,kC=2πR2iGataεx,po2ipoεx,pi1opiεx,F1oFεx,T1oTE44

To specify the boundary conditions of Eq. (43), one can consider the disappearance of F2at x = 0. This is because the left end surfaces of the adherend are normal traction-free. The right end at x = L on the other hand must take the full axial load F if there exists the application of external axial load. Thus, in the mathematical form, these boundary conditions are as follows:

F20=0,F2L=FE45

Finally, it should be noted that the occurrence of p2iin Eq. (43) is closely related with the existence of F2because the tensile or compressive loading can induce the peeling traction. Therefore, p2iin the equation is considered as unknown. However, it is possible to find the approximated relation between the two variables by letting T2and p2ibe zero; F2then can be evaluated and expressed in Eq. (46):

F2x=a0ekFx+b0ekFxkCkFE46

where a0and b0are integration constants.

Reinstating the resultant normal traction p2iand substituting Eq. (46) into Eq. (43), it is found that p2ican be simply estimated as

p2ia1ekFx+b1ekFxE47

in which, a1and b1are unknown parameters. In order to determine these two parameters, two more boundary conditions are required from zero longitudinal shear stress τxrain the adhesive layer at left and right ends as shown in Eq. (48):

dF2dxx=0=0dF2dxx=L=0E48

Up to this point, the unified formulation of an analysis of adhesive-bonded coupler joint has been developed. The model can be universally used to determine the stresses in the adhesive layer for any particular load case previously mentioned. To elaborate the applicability of the model for each loading condition, i.e., torsion, axial, or external and internal pressure, the pertinent details are given below:

  • For torsional load, the secondary variables, namely, F2and p2i,are initially neglected in Eq. (38). In addition, F=p1i=p2o=0. Consequently, T2and τθracan be evaluated. Subsequently, F2, τxra, and σraand can be recovered and computed by employing the full form of Eqs. (43) and (47), (28), and (35), respectively.

  • For axial load, the secondary variable T2is initially neglected. Additionally, T=p1i=p2o=0.The primary variables F2, τxra, and σraare solved by using the governing equation, Eqs. (43), (47), (28), and (35). T2and τθrais later calculated from the full form of Eqs. (38) and (23), respectively.

  • For external and internal pressure, the secondary variable T2is firstly omitted. In this case T=F=0.If only external pressure is present, p1i=0, whereas if only internal pressure exists, p2o=0. The first variables F2, τxra, and σraare solved by using the governing equation, Eqs. (43), (47), (28), and (35). T2and τθracan be later recovered the same way as those for the axial load.

3.3. Finite segment solution for evaluating adherend stresses

As previously discussed, all resultant loads in adherend parts 1 and 2 can be obtained as functions of x-coordinate. Equipped with the elasticity solution discussed in Section 2, the stress analysis in the adherends can be performed by utilizing a technique so-called finite-segment method (FSM) developed in [13]. The joint in the overlap region is divided into n numbers of segment as illustrated in Figure 3 to take care of the axial variation of the resultant loads. The corresponding resultant loads of each segment are then approximated to be constant, and thus, application of the theory delineated in the previous section is valid. For a composite coupler, the fiber angle in the adherend part 2 is denoted as Ø2. Note that if the stresses in adherend part 2 are evaluated, the internal pressure piin Eq. (21) or equivalently p2ishould be substituted by the radial normal stress σraobtained from Eq. (35) for more accurate results. Likewise, the procedure must be done for p1oif adherend part 1 is considered. When all of the resultant loads in Eq. (21) in each segment are completely determined, stresses in both adherends can be finally computed and analyzed.

Figure 3.

A segment at a certain axial position x.

4. Results

Presented in this section are some sample computational results of the model developed. The numerical calculation is performed by using software MATHEMATICA™. The validation of the model is not given herein, since it has already been shown in [13] and [20]. Adhesive-bonded tubular joints with isotropic inner adherend and symmetric-balanced four-layer stacking sequence ±Ø2scouplers are selected for consideration. The reference joint geometry is given with parameters R10=10, t1=5,t2=5,ta=0.1,andL=40mm. The adherend part 1 is made of steel, whereas the adherend part 2 is fabricated from carbon fiber-reinforced plastic. Epoxy is used as the adhesive material. The material properties of the joints are listed in Table 1. Adhesive normal stress ratio, α, is set to be 20 because it has shown to provide accurate predictions of adhesive radial stresses [20]. In the following computational results, the adhesive stresses in the coupler joint are normalized by the average applied stress in each loading case, because the dimensionless stresses are readily exploited to identify the level of load distribution intensity in the joint.

PropertiesEpoxy (adhesive)Steel (adherend 1)Carbon/epoxy (adherend 2)
E1(GPa)1.30200.00128.00
E2, E3(GPa)1.30200.0010.00
G12, G13(GPa)0.4676.9050.00
G23(GPa)0.4676.9050.00
υ12, υ130.410.300.28
υ230.410.300.47

Table 1.

Mechanical properties of materials in the principle material coordinate [20].

4.1. Torsional loading

In the case of torsional loading, the joints are assumed to have a torque of 1 N.m as an input without loss of generality. Also, the adhesive mean shear stress τmain Eq. (49) is utilized to normalize the induced adhesive hoop shear stress in the coupler joint:

τma=T2πR2i2LE49

The resultant torque of the adherend part 2 and normalized adhesive hoop shear stress can be calculated and plotted in Figures 4 and 5, respectively. It can be noticed that the joints considered develop the nonconstant slopes in Figure 4 with relatively high torque gradients at both ends. This is equivalent to the peak adhesive hoop shear stresses at x = 0 and L in Figure 5. Note that the torque and stress distributions for Ø2= 0°and 30°are identical to those for 90°and 60°, respectively, so they cannot be clearly seen. In addition, since the fiber orientation of 45°is the most suitable angle to withstand the in-plane shear loads, the coupler with Ø2= 45°provides the lowest magnitude of τθraas expected.

Figure 4.

Torque distribution in adherend part 2 of composite coupler joint subjected to torsion.

Figure 5.

Adhesive hoop shear stress distribution of composite coupler joint subjected to torsion.

Stress distributions in the composite coupler are illustrated in Figure 6 for the case of Ø2= 30o. The normal stress in the fiber direction σ11in Figure 6(a) is the dominant stress component compared to those in the other directions. The radial normal stress σ33illustrated in Figure 6(b) is relatively small at x = 0 mm and noticeably larger at x = 40 mm. In addition, σ33at adhesive-coupler interface r = 10.1 mm or σrais also minimal because of being the secondary effect. Note that the number of segments used to calculate the stresses in Figure 6 is 40. Finally, the figure shows that the developed model is capable of capturing the variation of these two stress components through the coupler thickness thanks to the advantage of the elasticity theory.

Figure 6.

Distribution of stresses in composite coupler subjected to torsion.

4.2. Axial loading

When the coupler joints are subjected to an axial loading, a tension force with the magnitude of 1 N is used in calculation. For this particular case, the adhesive mean shear stress τmain Eq. (50) is adopted to normalize the induced longitudinal shear stress and radial normal stress in the adhesive. Same as above, the normalized stresses can be utilized to indicate the distribution intensity of load transfer within the joints:

τma=F2πR2iLE50

Figures 7 and 8 show the effect of fiber orientation on the distributions of F2and τxraalong the overlap region, respectively. Observation in Figure 8 reveals that by adjusting fiber orientation, the composite coupler can generate mostly uniform load transmission in the central bonding region. The internal forces F2of Figure 7 in that region concomitantly reveal linear relationships with the spatial coordinate x/L. The optimum fiber angle Ø2is about 30°, which provides the lowest maximum τxra/τmaof 1.2. Figure 9 shows the radial normal stress in the adhesive. Generally speaking, the smaller peak of τxra/τma, the lower magnitude of σra/τma.

Figure 7.

Force distribution in adherend part 2 of composite coupler joint subjected to tension.

Figure 8.

Adhesive axial shear stress distribution of composite coupler joint subjected to tension.

Figure 9.

Interfacial radial stress distribution of composite coupler joint subjected to tension.

Figure 10 shows the normal stresses σ11and σ33of adherend part 2 in the principal material coordinate system, when the fiber orientation is equal to 30o. It can be seen from the figure that under the application of the axial force, σ11is vanished at the left of the bonding region due to the traction-free surface, while σ33is disappeared on the outermost area of the coupler. The stress in the fiber direction σ11steadily attains the same maximum value along the bond length in all laminae of composite coupler. The radial normal stress σ33, which is induced from the resultant axial force, is highest at the adhesive-coupler interface.

Figure 10.

Distribution of stresses in composite coupler subjected to tension.

4.3. Pressure loading

Lastly, for the case of pressure loads, 1 MPa internal pressure is exerted inside the adherend part 1, but no external pressure is present on the outer surface of the adherend part 2. The adhesive longitudinal shear stress and adhesive radial normal stress can then be normalized by the internal pressure to form the dimensionless variables.

Figures 11 and 12 show the effect of fiber orientation on the distributions of F2and τxraalong the overlap region, respectively. Figure 11 indicates that peak values of F2are generated in the central region of the composite couplers, but their values are null at both ends. The longitudinal shear stresses in adhesive τxrain Figure 12 illustrate the antisymmetric characteristic along the bond length. It can be seen that the optimum fiber angle Ø2is 90°. This fiber orientation delivers the lowest maximum τxra/piof 0.6. Figure 13 shows the radial normal stress in the adhesive. Interestingly, the values of σraare reduced by four to five times compared to the internal pressure applied over the whole range of the fiber angles considered.

Figure 11.

Force distribution in adherend part 2 of composite coupler joint subjected to internal pressure.

Figure 12.

Adhesive axial shear stress distribution of composite coupler joint subjected to internal pressure.

Figure 13.

Interfacial radial stress distribution of composite coupler joint subjected to internal pressure.

The normal stresses σ11and σ33of adherend part 2 in the principal material coordinate system, when the fiber orientation are equal to 30°, are displayed in Figure 14. It can be noticed that under the application of the uniform internal pressure with 1 MPa magnitude, σ11is maximum at the mid-length of bonding region, whereas σ33is peak at x = 0 and 40 mm on the adhesive-adherend interface.

Figure 14.

Distribution of stresses in composite coupler subjected to internal pressure.

5. Conclusions

A unified mathematical model for predicting the joint stresses of the adhesive-bonded tubular-coupler joints or the equivalent bonded-lap joints under several types of load is formulated. The inner and outer adherends can be considered as an isotropic material, orthotropic material, or a laminated composite, whose fiber angle is constant along the tube axis. They are modeled as three-dimensional body and satisfied the equilibrium, kinematic, and constitutive equations in theory of elasticity. The adhesive is only treated to be a very thin isotropic elastic material with relative low modulus, and thus, merely three out-of-plane stress components are present. The finite-segment method is developed to compute adherend stresses in each small portion of the coupler. The analytical results obtained indicate the viability of the model for many joint conditions and configurations. The model can be used conveniently in the preliminary process of the joint design, which is usually critical in huge, complex, or integrated structures.

© 2017 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

How to cite and reference

Link to this chapter Copy to clipboard

Cite this chapter Copy to clipboard

Sontipee Aimmanee (December 20th 2017). A Unified Analysis of Adhesive-Bonded Cylindrical Coupler Joints, Applied Adhesive Bonding in Science and Technology, Halil Özer, IntechOpen, DOI: 10.5772/intechopen.72288. Available from:

chapter statistics

421total chapter downloads

More statistics for editors and authors

Login to your personal dashboard for more detailed statistics on your publications.

Access personal reporting

Related Content

This Book

Applied Adhesive Bonding in Science and Technology

Edited by Halil Ozer

Next chapter

Development and Application of Low-Temperature Curable Isotropic Conductive Adhesive Toward to Fabrication in IoT Generation

By Yusuke Okabe

Related Book

First chapter

Finite Element Analysis in Dental Medicine

By Josipa Borcic and Alen Braut

We are IntechOpen, the world's leading publisher of Open Access books. Built by scientists, for scientists. Our readership spans scientists, professors, researchers, librarians, and students, as well as business professionals. We share our knowledge and peer-reveiwed research papers with libraries, scientific and engineering societies, and also work with corporate R&D departments and government entities.

More About Us