The Hybrid Joints between an FRP Panel and a Steel Panel through Tubular Reinforcements: A Methodology for Interlaminar Stress Calculations

: The advantages of laminates in terms of the chemical properties and mechanical properties/weight relationship have motivated several applications of fiber-reinforced plastic (FRP) composites in naval constructions due to the reduction in structural weight. This weight advantage has motivated multiple investigations dedicated to dissimilar material joints. We present a methodology for the interlaminar stress calculations of a tubular hybrid joint between an FRP panel and a steel panel through tubular reinforcements. The proposed formulas allow the estimation of the shear and normal stresses on the adhesive, which are generated in the bonding angle of the tubular hybrid joint. The stresses generated at the adhesive bonding ends influence on the adherent’s adjacent layer. A failure criterion is shown to check the accomplishment of the resulting stresses in the adherent laminate. Finally, the proposed formulas are validated using the finite element method and compared with the obtained interlaminar stresses. indicating the typical interlaminar failures in a hybrid joint with tubular reinforcement.


Introduction
The hybrid joint was developed from the technique of using adhesive and bolts to bond different materials, which are applied to join steel deck and fiber-reinforced plastic (FRP) panels. The different proposals of many researchers are summarized as follows: Ritter and Speth [1] developed a proposal for a hybrid union of a military ship, between a steel deck and composite superstructure with symmetrical connection is made using an adhesive double joint. This joint was proven by converting two German frigates into yachts. Rudiger and McGeorge [2] developed the symmetrical joint, an adhesive double type with a sandwich panel and without a bevel in connection. The joint was fixed using adhesive and supported by a mechanical fastening.
Hentinen and Hildebrand [3,4] developed an asymmetric single-type adhesive joint with a sandwich panel and transition bevel; the critical load of the joint was absorbed by the bolts. Bohlmann and Fogarty [5] proposed a hybrid joint with bolts, adhesive, and a steel plate that could be welded directly to the deck and screwed or joined to a sandwich-type laminate. Kotsidis, Kouloukouras, and Tsouvalis [6] analyzed the finite elements of a sandwich-type joint, but with a modification, including [33], and Huang et al. [34], allowing them to estimate the S-N curve of the joint based on the operating cycles.
In this investigation, the methodology consists of using general formulas to calculate the stresses of adhesive joints applied in a tubular hybrid joint (see Figure 1), and the accomplishment of a failure criterion of interlaminar stresses in the laminates of the adherents. From the different tests performed with finite element analysis (FEA), it was proved that the tubular reinforcement laminate behaves like an adhesive single-lap joint. Bigwood and Crocombe [24] formulas are used due to different thicknesses and stiffness of the adherents of this joint. These formulas have been validated by Nhamoinesu and Overend [35] and Riccio and Sellitto [36] using FEA. Nhamoinesu [35] applied the method to select the epoxy and acrylate adhesive based on the mechanical performance of the bond and subjected to short-duration loads; this analytical method has shown good results with small deformations in the adhesive (30% of the maximum deformation). However, it loses accuracy with greater deformation compared with the nonlinear FEA results. Riccio [36] apply the Bigwood [24] method as it is of general use in adhesive joints; they used an L-shaped symmetric T-pull model modelled in FEA with four nodes 2D plane elements, finding satisfactory results in the extreme stress values and the distribution along with the adhesive.
From the results of the different studies, we found that depending on the characteristics of the adhesive joint, the adhesive or the interface between the FRP laminates adherents 1 and 2 may fail [18]. For this reason, when studying a hybrid joint using tubular stiffeners, consideration should be given to the following: 1. The limit of the interlaminar stresses of the adherents, Hou [37]. 2. The fatigue on the joint to guarantee its useful life, DNV [32]. For the study of interlaminar stress, the following tubular hybrid joint configuration is considered: • Adherent 1: fiberglass laminated over the metal pipe. • Adherent 2: a sandwich panel FRP.
The orthotropic properties for adherent laminates 1 and 2, are in Appendix A: (Table A1-A3); the formulas to calculate the 3D stress-strain matrix, the transversal properties of the individual ply, the equivalent isotropic properties, are in Appendix B.

Methodology
The proposed methodology is used to obtain the resulting stresses at the ends of the adhesive based on the equivalent isotropic properties of adherents 1 and 2. The simplified formulas proposed by Bigwood/Crocombe [24] to calculate the stresses in the adhesive has the limitation that does not consider the coupling between normal and shear stresses, due to this limitation, the results are useful for a preliminary analysis of the tubular hybrid joint. Second, the interlaminar stress analysis in the adherent's adjacent outer layer is carried out, because the properties of the individual ply of the adjacent layers are considered. This part is a localized microstructural analysis to study laminate debonding as a result of fatigue, this debonding is avoided when the quadratic-type failure criterion is met.

Proposed Formulas for Tubular Hybrid Joint
In this section, a mathematical model concerning the behavior of tubular reinforcement on an FRP panel is formulated. For the analysis, we considered that the reinforcement in the panel remains attached by the adhesive.

Macro-Structural and Microstructural Analysis
The macro-structural analysis consisted of considering the composite laminate as a monolithictype solid-elastic body, and allowed a preliminary estimation, based on the following considerations, [38]: • The structural laminate has a linear behavior; therefore, it complies with Hook's Law [18]. • Transversal deformations due to shear stress are negligible.
• Deformations vary linearly across the thickness.
• The unitary elongation in the vertical direction is considered as being negligible compared to the unitary elongations of the horizontal plane.
• The laminate is quasi-isotropic.
• The formulas for macro-structural analysis are based on the classical theory of plates. The structural laminate panel is considered as being circumscribed and fixed between girders.
The microstructural analysis considered that the layers of the laminate can be approximated with a rectangular laminate stack. This allowed the final estimation of the stresses and deformations of each laminate layer. This type of analysis is also known as a 'critical layer' study.

Adhesive Model Considerations
The normal and shear stresses of the adhesive model had the following considerations: • The Bigwood/Crocombe [24] model considers the adhesive as isotropic, elastic, and linear; • The adherent's properties are considered isotropic and can be of different thicknesses; • The adhesive used is polyester resin with the same characteristics of those considered in the Bigwood/Crocombe model [24]; • The adhesive joint of the panel tubular reinforcement is analyzed according to the theory of elasticity; • A typical diagram of the joint between the reinforcement and the panel is shown in Figure 2;

Equations of Motion in terms of normal and shear stress, simplified analysis
For normal stress, the equation of motion is simplified, based on the consideration that the variation of the shear stress along the joint is lesser in importance [24]. where: using the following compliance factors for the normal stress calculation: Based on compliance of these two factors (Equations (3) and (4)), the following formulas are obtained to determine the maximum values of normal stresses: For the shear stress, the following motion equation is simplified, based on the assumption that the variation of the normal stress along the joint is small. where: using the following compliance factors for the calculation of shear stress: Based on the compliance of these two factors (Equations (9) and (10)), the following formulas are obtained to determine the maximum values of shear stresses: The definition of the variables Vc, Tc, and Mc are indicated in Figure 2. These formulas can be applied to two types of panels the single laminate and the sandwich laminate. Following the model considerations indicated in Section 2.1.2, we expected to estimate the stress to occur in the adhesive.

Forces and Moments on the Tubular Hybrid Joint
The elastic model of Bigwood/Crocombe [24] satisfied the model considerations proposed in Section 2.1.2 and allows us to estimate the adhesive stress of the tubular hybrid joint. To apply these simplified formulas in the tubular hybrid joint, it is necessary to define the forces and moments, as explained below.
In Figure 3, we considered that the reinforcement is subjected to forces (P, V, T) in three directions. In this study, the forces used in the formulas proposed by Bigwood/Crocombe [24] are applied per unit length (see Section 2.1.2 and Appendix C). Where: • P is the force in the y-direction on the hybrid reinforcement; this force generates an equivalent moment M' exerted in the tubular laminate. This moment is estimated as follows (see Figure 4): Section view FRP Panel • V is the force on the hybrid reinforcement in the z-direction. This force is broken into its two components (Vz, Vy), separated by the pipe diameter D. See Error! Reference source not found.a, on both sides of the tubular laminate. The components are defined as follows: The component (Vy) generates a moment in Adherent 1. The behavior of the tubular laminate in the y-direction is similar to that proposed by Hart-Smith's moment equation [18], see Error! Reference source not found.b: in which: • T is the force applied to the tubular laminate in the x-direction; this force is applied to both the laminates and the root to obtain the adhesive shear stress.

Application to Tubular Hybrid Joint
A typical application of the hybrid tubular joint between a steel deck and FRP panel of a superstructure is presented in Error! Reference source not found.. The proposed methodology allows estimating the adhesive stresses, so it is necessary to define the forces (P, V, T), these forces must be calculated in advance to be applied over a tubular hybrid joint of the FRP panel, under pressure p (kN/m 2 ).
To estimate the resultant forces (P, V, T) on the pipes of the hybrid joint, a global analysis of a superstructure subjected to a pressure p is performed, additionally considering a heeling or pitching. The magnitude of p and the heel or pitch is recommended by the classification society.
In cases where the hybrid joint is installed on the steel deck where there is auxiliary equipment, the moments transferred from the deck to the panel should be added in calculations.

Failure Criteria
Once the shear and normal adhesive stresses resulting from the applied forces are obtained with equivalent isotropic properties, it is important to evaluate the combination of the interlaminate stresses of the adherent's adjacent outer layer with the properties of the individual ply. The stresses resulting from the Bigwood/Crocombe [24] formulas are considered decoupled by definition. Therefore, the quadratic criterion of Tsai-Hou [37], Equation (17), is selected as the failure criterion.
When designing a panel with stiffeners, the type of load to be applied or the fatigue to which it is being subjected must be considered. For this reason, when evaluating the quadratic failure criterion (Equation (17)), the calculated static stresses must be reconsidered as virtual stresses, which includes the type of load to which they are being subjected. These virtual stresses are defined as follows [33]: = . (18c) The modified failure criterion is proposed considering the virtual stresses as indicated: where: According to Equations (5), (6), and (11)-(13) applied to the tubular hybrid joint, we considered the following: The formulas indicated are based on linear theory; therefore, the principle of stress superposition can be applied when the resulting stresses are in the same direction. To carry out the optimization process, we considered that not only the first layer must comply with the failure criteria, but all the layers of the laminate including the core must also comply if this is applicable. For this, a matrix must be generated to estimate the behavior of each laminate layer. To estimate the stresses in the successive layers, the formulas proposed by Special Service Craft (SSC) Lloyd's Register [39] can be used to verify that the failure criterion is met.

Limit Interlaminar Stresses
The normal and shear stresses' limits in an interlaminate depend on the amount of fiber and the type of resin used. Table 1 shows the limit values of the interlaminar stresses for a laminate with an orthophthalic polyester resin. In the case where one of these parameters is not available, it can be estimated using the following dimensionless relationship (Equation (21b)). For cases in which the FRP panel is of sandwich-type, with a balsa core, additional consideration must be given to the tensile, compression, and shear stresses of the core, see Lloyd´s Register [39]: Table 2.2.2 Minimum characteristics and mechanical properties of end-grain balsa.

Design Factors: Interlaminar Stress Fraction
When designing an adhesive hybrid joint, the type of force that the panel is subjected to must be considered to estimate the life of the hybrid joint. Hollaway [33] recommended using the fractions of interlaminated stress shown in Table 3 for the design of a hybrid joint, depending on the life cycle, see Hollaway [33]: Table 5.2 Typical long term load reduction factors for GFRP: data refer to the ratio of long term strength/short term strength at ambient test conditions. As a reference, Lloyd's Register [39] Figure 3 shows the model used for the validation of a tubular hybrid joint, which consists of a 600 × 210 × 32.9 mm FRP panel, a 2" Sch40 steel pipe, and a tubular laminate with four layers of fiberglass over the pipe, the laminate sequence stacking and properties are in Table A2. To apply the formulas in Section 2.1.2 and Appendix C, the equivalent isotropic properties must be estimated from the orthotropic properties (see Appendix B).

Numerical Model
For comparison of results, finite element analysis is performed; 2D and 3D models are developed using Ansys software, considering the orthotropic properties of the laminate in each adherent. A 0.1 mm polyester resin interface layer is modelled between each layer of the laminates. The 3D model is analyzed with the ACP-Ansys module since it allows entering the properties of the adherents per layer; the adherents are modelled using shell elements and the adhesive using solid elements.
For the 2D model, only the Static Structural module with plain strain configuration is used since it allows analyzing the behavior of laminates and interfaces in the tubular hybrid joint. Software tools for mesh quality analysis are used and the convergence of the stress results with the indicated mesh for the 2D and 3D models, is achieved using a mesh refinement as indicated below:   The force is applied, taking into consideration the entire laminate surface of the reinforcement excluding the bonding tubular laminate over the panel.

Results
The following Figures 8-10 present the comparison of stresses obtained from 2D, 3D FEA models and the general formulas of Bigwood and Crocombe [24] presented in Appendix C. The stress curves of formulas [24] show similar behaviors to the results obtained with the numerical analysis (FEA), however, the normal stress generated by the T force (Figure 5a) is zero.   The distribution of the maximum shear stresses of the T force, from the 3D model, at the first interface stress distribution between layers 1 and 2 of adherent 1 is presented in Figure 8. The distribution of normal and shear stresses on adhesive and adherent 1 interfaces resulting from the V force, 2D model, are presented in Figure 9. The results obtained in the adhesive at the endpoints A and B (Figure 3) with the proposed normal and shear stress formulas, and stresses calculate with FEA are presented in Table 2 and 3 corresponding to the forces P, V, and T. Only results are presented in points A and B because they have greater values than points C and D. These tables present the application of the failure criterion (Equation (19b)), showing that the criterion is met. Using the formulas of the failure criterion it is possible to obtain the limit values of the safety factor and the design pressure p that the adhesive joint can withstand. Table 4 presents the limit design pressure p, keeping constant interlaminar stress fraction at 0.650. Similarly, Table 5 presents as an alternative to the maximum safety factor to which this FRP panel can be subjected, maintaining a constant design pressure of 5.160 kN/m 2 . These indicated limit values of p and Sf application are subject to compliance of the critical layer analysis of FRP panel proposed by SSC-Lloyd's Register, [39].
The safety factor Sf is defined as:

Discussion
The method proposed by Bigwood and Crocombe includes simplified and general formulas that allow knowing the maximum and distributed values of the normal and shear stresses along with the adhesive. The results obtained with both formulations result in similar values and simplified formulas allow the quick estimation and easy application.
2D and 3D models have the same boundary conditions, making it possible to make a comparison between the results of the stresses. The 2D model is made for a local analysis allowing a more refined mesh; the stress distribution results obtained demonstrate a slight difference in their extreme values of the adhesive, attributed to boundary singularities of the model. Figure 5a shows that applying a horizontal force (T), the Bigwood and Crocombe formulas do not calculate the normal stress, however, the FEA model shows the normal stress.
Regarding the points defined in Figure 3, it can be shown in Figure 7 that when the force V is applied at point A, there is a shear stress value of 1.031 MPa, while at the extreme point B the shear stress is 3.076 MPa, this difference in stress is due to the moment reaction generated at the boundary condition in the adherent 2.
In Figures 11 and 12, the behavior of the stresses through the laminate and its interfaces is shown. The stresses of the adhesive are like those presented in the first interface of the adherents 1 and 2, therefore the thickness of the interface and the adhesive of the joint must be considered.
In Figure 9 of the 2D model, high-stress values are presented in the corner of the external laminate of the adherent 1 due to the stress concentration in this corner.
A tubular hybrid joint (see Error! Reference source not found.) can be optimized based on the number of layers of the tubular laminate, configuration of the tubular structure, and the limit pressure that the FRP panel and the hybrid tubular joint can support. In cases where it is required to increase the pressure p, the FRP panel laminate should be reviewed utilizing critical layer analysis, and as an option closer vertical pipes can be installed.

Conclusions
Initially, for the preliminary design, the macro-structural analysis should be carried out on the scantling of the panel structure, using a Classification Society guideline. Then, a microstructural analysis of the laminate sequence should be carried out using a "critical layer" analysis. Finally, the hybrid tubular joint is dimensioned as an additional reinforcement of the panel, according to the forces acting on the hybrid joint.
The number of laminate layers on the tubular reinforcements depends on the type of design load that are applied or transmitted to the panel and/or tubular hybrid joint; thus, the laminate scantling should be validated by the failure criteria using the interlaminar stresses.
The model considerations are applicable for the interlaminar study but do not include the calculation of the temperature or hygroscopic effects. Another aspect is the tubular laminate bevelling, as the maximum resulting stresses are located at the reinforced laminate ends. For this reason, we recommend that the bevel guide proposed by SSC-Lloyd's Register be applied.
The proposed moment formula is recommended for tubular reinforcement. In the case where it is necessary to apply another structural reinforcement type, the formula must be modified, the compatibility and stresses between laminates should be analyzed.
The interlaminar stresses resulting from the outer layer are related to the possibility of debonding; however, in the case of sandwich panels as a second stage, we recommend analyzing the interlaminar and intralaminar stresses of all layers and the core. In no case, the core should be used perpendicular to the stress direction.
When comparing the interlaminar stress fraction recommended by SSC-Lloyd's Register with that proposed by Hollaway, this fraction corresponds to 10 6 lifecycles, assuming that the core tie layer is CSM. Additionally, Hollaway proposed that the fraction may also have lower values than those recommended by SSC-Lloyd's Register, depending on the fatigue applied. For this reason, we recommend researching the variation of the interlaminar stress fraction as a function of fatigue.
The formulas proposed for the shear and normal stress estimation showed satisfactory results when compared with the values obtained with FEA. The proposed methodology made possible the study of the behavior of the stresses obtained at the adhesive ends and the analysis of the resulting interlaminar stresses that are generated in the adherent's adjacent outer layer. The stress analysis in of the remaining layers and the interlaminar effect on the core, is beyond the scope of this investigation, however, is an important topic of study in the future. For the proposed equations, the failure stresses for laminates of polyester resin with E fiberglass (FRP) are presented in Table 1.
The proposed methodology is aimed to facilitate the preliminary design of a tubular hybrid joint. The joint as designed allows to join the FRP panel with metal decks having the advantage of getting a watertight union, improving the contact area of the adhesive, and obtaining an inert hybrid metal structure, having as an additional advantage that, depending on the type of load condition on the FRP panel and steel deck, it is easy to increase the number of vertical reinforcements or the adhesive joint length.

Conflicts of Interest:
The authors declare that they have no known competing for financial interests or personal relationships that could have appeared to have influenced the work reported in this paper. (2) Laminate 2: CSM300 + CSM450 + WR800 + CSM450 + Balsa 25mm + CSM450 + WR800 + CSM450 + CSM300. Table A3 presents the equivalent isotropic properties resulting from the orthotropic laminate of adherents 1 and 2 (see Table A2), using the formulas indicated in Appendix B.1.3. This equivalence is necessary to apply the Bigwood and Crocombe [24] formulas indicated in Section 2.1.2 and Appendix C. The presented formulas correspond to the stress-strain matrix according to the classical lamination theory. Parameters are defined on Vinson [38]. Properties of orthotropic laminates are defined by Naughton [40]: generalized formulas. (A10b)