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

Abstract

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.

1. 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 the inner end of a folded steel plate. Shkolnikov [7] proposed a hybrid joint between steel panels and FRP panels; this union has been called ‘Comeld-2′. This study focused on the application of composite panels in military and submarine ships. Simler and Brown [8] proposed a US Navy ONR’s Navy Joining Center (NJC) adhesive joint consisting of a metal H profile to generate an adhesive double joint, which is to be welded to the deck. The lamination used was vinyl ester resin and a balsa core.

Boyd, Blake, Shenoi, and Kapadia [9] performed a local fatigue analysis on a composite-steel panel, between the joint of the deck and the side of a frigate superstructure. Ozes and Neser [10] conducted experimental tests of hybrid joints between steel and fiber-reinforced panels, using the infusion method, for different roughness states of the steel. Babazadeh and Khedmati [11], in their publication, analyzed the effect of the main parameters of the hybrid joint, sandwich-type with adhesive, through applying tension loads. Johansson [12] and Beeston [13] studied the type of construction used by the Kockums shipyard for the joints of reinforced fiberglass/carbon fiber reinforced polymer (CFRP) superstructures with steel hulls. The panel contact was made on the U profile through adhesives.

The mentioned hybrid joints were applied mainly to join FRP superstructures, masts, and accessories to a metal structure. The proposed joints were mostly of the adhesive type, and a few involved fixing bolts. The methods proposed had some limitations as follows: the difference between the thermal expansion coefficients; the corrosive effect if the steel structure was used; the difficulty of fixing the FRP panel girders to the metal structure; and the fixing of the interface structure to the metal panel. In the present investigation, we used a tubular hybrid joint, as proposed in [14,15], as an option that allowed the following: prefabricating the FRP block, minimizing the difference in the thermal expansion coefficients, and minimizing the corrosive effect due to the metal surfaces that are protected with an adequate epoxy anticorrosive treatment.

Researchers proposed different techniques for the analytical study of adhesive single-lap joints; these methods can be classified as explicit and implicit. Explicit methods to estimate shear and normal stresses (closed form) are as follows: Volkersen [16] was the first to study single adhesive joints, he only proposes the shear stress formula because it does not consider deformations in the adherents. Goland and Reissner [17] propose formulas considering the deformation in the adherents due to the bending moment generated by the eccentricity of the applied force, its method only applies to balanced joints and thin adherents. Hart-Smith [18,19,20,21,22] developed formulas for the stresses considering elastic and plastic adhesive material for balanced and unbalanced joints. Allman [23] developed the formulas for balanced joints considering a non-deformable geometry, concluding that the shear stresses do not vary along with the adhesive. Bigwood and Crocombe [24,25] investigates the shear and normal stresses applicable to various adhesive joint configurations and different adherents, with simplified formulas that are easy to apply. Oplinger [26] following the work of Goland and Reissner, modified the formulas to estimate the stress of the adhesive, considering the thickness of the adhesive and different thicknesses of the adherents. Zou [27] proposed easy-to-apply formulas for estimating adhesive joints only for balanced adherents. These explicit methods are applied for adherents and adhesives with isotropic properties and linear-elastic analysis.

Implicit methods include the following: Renton and Vinson [28] who developed an ordinary linear differential equation of the eighth order, where they consider anisotropic adherents and isotropic adhesive. Ojalvo [29], based on the Goland and Reissner method, analyzed the influence of the thickness of the adhesive on the adhesive joint. Adam and Mallick [30] included the influence of thermal loads on the adhesive bond and the modelling of the spring-loaded adhesive and presented formulas for the stresses on the top and bottom of the adhesive. Smeltzer [31] proposed formulas considering elastic anisotropic adherents and elastic-plastic isotropic adhesive. These methods recommend using numerical methods and/or finite elements depending on the case to complete the estimation of the stresses of the adhesive joints.

The mentioned methods have proposed formulas to statically estimate the stresses and their distribution in the adhesive, however, they have not taken into consideration the behavior of the joint subjected to vibratory loads. These studies are presented by Det Norske Veritas (DNV) [32], Hollaway [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:

• The limit of the interlaminar stresses of the adherents, Hou [37].
• 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.
• Adhesive: a 250-micron polyester resin layer.

The orthotropic properties for adherent laminates 1 and 2, are in Appendix A: (

Table A1. Fibers, resin, and core properties for individual ply.

Properties	Symbol	CSM 300	CSM 450	WR 800	Balsa	Polyester Resin
Density (kg/m3)	ρ	1384	1428	1634	144	1200
Elasticity modulus x-direction (N/mm2)	$E_x{}^{i-p}$	5750	6500	14,500	67.80	3400
Elasticity modulus y-direction (N/mm2)	$E_y{}^{i-p}$	5750	6500	14,500	67.80	3400
Elasticity modulus z-direction (N/mm2)	$E_z{}^{i-p}$	4169	4414	6017	3900	3400
Shear modulus xy-direction (N/mm2)	$G_{xy}{}^{i-p}$	2665	2750	3090	19	1250
Shear modulus xz-direction (N/mm2)	$G_{xz}{}^{i-p}$	1607	1713	2364	129	1250
Shear modulus yz-direction (N/mm2)	$G_{yz}{}^{i-p}$	1607	1713	2364	129	1250
Poisson coefficient xy	${\mu }_{xy}{}^{i-p}$	0.340	0.340	0.320	0.285	0.360
Poisson coefficient xz	${\mu }_{xz}{}^{i-p}$	0.340	0.330	0.300	0.014	0.360
Poisson coefficient yz	${\mu }_{yz}{}^{i-p}$	0.340	0.330	0.300	0.014	0.360

,

Table A2. Orthotropic properties of adherents.

Properties	Symbol	Adherent 1 (1)	Adherent 2 (2)
Density (kg/m3)	ρ	1513	499
Thickness (mm)	h	4.030	32.900
Elasticity modulus x-direction (N/mm2)	$E_x$	8569	2064
Elasticity modulus y-direction (N/mm2)	$E_y$	8569	2064
Elasticity modulus z-direction (N/mm2)	$E_z$	5119	4249
Shear modulus xy-direction (N/mm2)	$G_{xy}$	2713	652
Shear modulus xz-direction (N/mm2)	$G_{xz}$	1770	166
Shear modulus yz-direction (N/mm2)	$G_{yz}$	1770	166
Poisson coefficient xy	${\mu }_{xy}$	0.421	0.421
Poisson coefficient xz	${\mu }_{xz}$	0.490	0.170
Poisson coefficient yz	${\mu }_{yz}$	0.490	0.170
Poisson coefficient yx	${\mu }_{yx}$	0.421	0.421
Poisson coefficient zx	${\mu }_{zx}$	0.293	0.349
Poisson coefficient zy	${\mu }_{zy}$	0.293	0.349

Note: (1) Laminate 1: CSM450 + WR800 + CSM450 + CSM450. (2) Laminate 2: CSM300 + CSM450 + WR800 + CSM450 + Balsa 25mm + CSM450 + WR800 + CSM450 + CSM300.

and

Table A3. Equivalent isotropic properties of adherents.

Properties	Symbol	Adherent 1	Adherent 2
Elasticity modulus (N/mm2)	$E$	6171	2157
Shear modulus (N/mm2)	$G$	2151	730
Poisson coefficient	$\mu$	0.435	0.477

); the formulas to calculate the 3D stress-strain matrix, the transversal properties of the individual ply, the equivalent isotropic properties, are in Appendix B.

2. 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.

2.1. 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.

2.1.1. Macro-Structural and Microstructural Analysis

The macro-structural analysis consisted of considering the composite laminate as a monolithic-type 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.

2.1.2. 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;

• The distance L of the joint length;
• The equations of motion are obtained assuming that the panel has a cylindrical flexure behavior.

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].

$$\frac{d^4{\sigma }_z}{\partial y^4}+4K_5{}^4{\sigma }_z=0,$$

(1)

where:

$$K_5{}^4=\frac{E_a}{4\cdot t}\left(\frac{1}{D_1}+\frac{1}{D_2}\right),$$

(2)

using the following compliance factors for the normal stress calculation:

$${\beta }_1=\frac{12\cdot E_a\cdot \left(1-{\mu }_1{}^2\right)}{E_1\cdot h_1{}^3\cdot t},$$

(3)

$${\beta }_2=\frac{12\cdot E_a\cdot \left(1-{\mu }_2{}^2\right)}{E_2\cdot h_2{}^3\cdot t}.$$

(4)

Based on compliance of these two factors (Equations (3) and (4)), the following formulas are obtained to determine the maximum values of normal stresses:

$${\sigma }_{V_c}=\frac{-{(2)}^{0.5}\cdot {\beta }_1\cdot V_c}{{({\beta }_1+{\beta }_2)}^{0.75}},$$

(5)

$${\sigma }_{M_c}=\frac{-{\beta }_1\cdot M_c}{{({\beta }_1+{\beta }_2)}^{0.5}}.$$

(6)

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.

$$\frac{d^3{\tau }_{yz}}{\partial y^3}-K_6{}^2\frac{d{\tau }_{yz}}{\partial y}=0,$$

(7)

where:

$$K_6{}^2=\frac{4\cdot G_a}{t}\left(\frac{\left(1-{\mu }_1{}^2\right)}{E_1\cdot h_1}+\frac{\left(1-{\mu }_2{}^2\right)}{E_2\cdot h_2}\right),$$

(8)

using the following compliance factors for the calculation of shear stress:

$${\alpha }_1=\frac{G_a\cdot \left(1-{\mu }_1{}^2\right)}{E_1\cdot h_1\cdot t},$$

(9)

$${\alpha }_2=\frac{G_a\cdot \left(1-{\mu }_2{}^2\right)}{E_2\cdot h_2\cdot t}.$$

(10)

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:

$${\tau }_{T_c}=\frac{-{\alpha }_1\cdot T_c}{2\cdot {({\alpha }_1+{\alpha }_2)}^{0.5}},$$

(11)

$${\tau }_{V_c}=\frac{3\cdot V_c}{4\cdot h_1},$$

(12)

$${\tau }_{M_c}=\frac{3\cdot {\alpha }_1\cdot M_c}{h_1\cdot {({\alpha }_1+{\alpha }_2)}^{0.5}}.$$

(13)

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.

2.2. 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):

$$M^{\prime }=\frac{M}{6}=\frac{P\cdot D}{12}.$$

(14)

• V is the force on the hybrid reinforcement in the z-direction. This force is broken into its two components (V_z, V_y), separated by the pipe diameter D. See Figure 5a, on both sides of the tubular laminate. The components are defined as follows:

$$V_z=\frac{V}{2},V_y=\frac{V}{2}\cdot \tan (30).$$

(15)

The component (V_y) 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 Figure 5b:

$$M_{H-S}=V_y\cdot d_1\cdot k_1,$$

(16)

in which:

$$k_1=\frac{1}{1+{\xi }_1\cdot c+\frac{{({\xi }_1\cdot c)}^2}{6}},$$

(16a)

$${\xi }_1{}^2=\frac{12\cdot V_y\cdot \left(1-{\mu }_1{}^2\right)}{k_b\cdot E_1\cdot h_1{}^3}.$$

(16b)

• 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 Figure 6. 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/m²).

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.

2.3. Interlaminar Stresses

2.3.1. 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.

$${\sigma }_z\ge 0,{\left(\frac{{\sigma }_z}{Z_t}\right)}^2+\left(\frac{{\tau }_{xz}{}^2+{\tau }_{yz}{}^2}{S_t{}^2}\right)\ge 1.$$

(17)

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]:

$${\sigma }_z{}^{\prime }=\frac{{\sigma }_z}{F_{rc-i}},$$

(18a)

$${\tau }_{xz}{}^{\prime }=\frac{{\tau }_{xz}}{F_{rc-i}},$$

(18b)

$${\tau }_{yz}{}^{\prime }=\frac{{\tau }_{yz}}{F_{rc-i}}.$$

(18c)

The modified failure criterion is proposed considering the virtual stresses as indicated:

$${\sigma }_z\ge 0,{\left(\frac{{\sigma }_z{}^{\prime }}{Z_t}\right)}^2+\left(\frac{{\left({\tau }_{xz}{}^{\prime }\right)}^2+{\left({\tau }_{yz}{}^{\prime }\right)}^2}{S_t{}^2}\right)\ge 1$$

(19a)

where:

$${\sigma }_z\ge 0,{\left(\frac{{\sigma }_z}{F_{rc-i}\cdot Z_t}\right)}^2+\left(\frac{{\left({\tau }_{xz}\right)}^2+{\left({\tau }_{yz}\right)}^2}{F_{rc-i}{}^2\cdot S_t{}^2}\right)\ge 1.$$

(19b)

According to Equations (5), (6), and (11)–(13) applied to the tubular hybrid joint, we considered the following:

$${\sigma }_z={\sigma }_{M_c\_P}+{\sigma }_{V_c\_V_z},$$

(20a)

$${\tau }_{xz}={\tau }_{T_c\_T},$$

(20b)

$${\tau }_{yz}={\tau }_{M_c\_P}+{\tau }_{V_c\_V_y}.$$

(20c)

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.

2.3.2. 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. Estimation of the failure stresses in the interlaminate.

Fiber	fc (1)	St(1)	Zt
		(MPa)	(MPa)
Uni-directional (UD)	60%	15.00	14.60 (1)
Woven Roving (WR)	50%	15.25	11.70 (3)
Chopped strand mat (CSM)	30%	17.95	9.50 (3)

(1) Lloyd’s Register [39], (3) Zt estimated using Equation (21b). Note: see laminate properties (E_mj y E_ij), in Appendix A.

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)).

$$\frac{Z_{t_1}}{S_{t_1}}\cdot {\left(\frac{E_m{}_1}{E_{i1}}\right)}^{1/2}=\frac{Z_{t_2}}{S_{t_2}}\cdot {\left(\frac{E_{m2}}{E_{i2}}\right)}^{1/2}$$

(21a)

$$Z_{t_2}=Z_{t_1}\cdot \frac{S_t{}_2}{S_t{}_1}\cdot {\left(\frac{E_m{}_1}{E_{m2}}\right)}^{1/2}\cdot {\left(\frac{E_{i2}}{E_{i1}}\right)}^{1/2}$$

(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.

2.3.3. 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 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] recommended using the failure stress fraction for tension or compression stresses [0.3–0.4], and shear stresses [0.33], according to the application area.

3. Results: Single-Lap Joint Validation by Applying FEA

The model of a cruising yacht, 36.80 m, proposed in [14] is used for this validation, which consists of applying a design pressure of 5.16 kN/m² on the side of the first port side deck with an area of 48.78 m² and a heeling of 30° to starboard, without considering any moment transmitted from the deck. Resultant forces on an intermediate tubular connection between the omegas are (see Figure 6) P = 134 [N], V = 5127 [N], y T = 4899 [N].

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).

3.1. 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:

• Software: Ansys Academic;

  ○

  Module: ACP Pre, ACP Post and Static Structural;

• Meshing 3D: Model size 600 mm, general mesh 4 mm, refinement to 1.5 mm on a 30 mm radius on pipe corners, and to 0.2 mm on the middle panel on a 10 mm radius. The adhesive has three elements through-thickness, (see Figure 7a);

  ○

  Nodes: 69,367;

  ○

  Element: Quadrilateral Dominant;

• Meshing 2D: Model size 210 mm, general mesh 0.7 mm and refinement to 0.15 mm on a 6 mm radius on adhesive ends. The adhesive has two elements through-thickness, (see Figure 7b);

  ○

  Nodes: 99,482;

  ○

  Element: Quadrilateral Dominant.

• Modeling: The metal pipe is not considered in the validation.
• Boundary conditions: The panel is considered as being simply supported in the (x, y, z) directions, at the ends of the panel (adherent 2). The tubular laminate (adherent 1) is considered as being free at its ends.
• Forces:

  ○

  For the x-direction (T):

  ▪

  The force is applied, taking into consideration the entire laminate surface of the reinforcement excluding the bonding tubular laminate over the panel.

  ○

  For the y-direction (P):

  ▪

  The force is applied on the side of the tubular laminate at a remote point in the center of the metal pipe.

  ○

  For the z-direction (V):

  ▪

  The force is applied, taking into consideration the entire laminate surface of the reinforcement excluding the bonding tubular laminate over the panel.

3.2. Results

The following Figure 8, Figure 9 and Figure 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 8a) 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 11.

The distribution of normal and shear stresses on adhesive and adherent 1 interfaces resulting from the V force, 2D model, are presented in Figure 12.

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. Failure criteria the tubular hybrid joint: Bigwood and Crocombe formulas.

Variable	Value	Bigwood and Crocombe Formulas
		A (0,0,0) mm			B (0,75,0) mm
		$\mathit{\sigma }$	$\mathit{\tau }$		$\mathit{\sigma }$	$\mathit{\tau }$
		(MPa)	(MPa)		(MPa)	(MPa)
		z	yz	xz	z	yz	xz
T (N/mm)	4899	0.000	-	7.858	0.000	-	7.858
V (N/mm)	5127	2.148	1.031	-	−0.652	3.076	-
P (N/mm)	134	0.509	−0.287	-	−0.003	0.016	-
${\sigma }_z$ (MPa)		2.657	-	-	−0.655	-	-
${\tau }_{yz}$ (MPa)		-	0.744	-	-	3.092	-
${\tau }_{xz}$ (MPa)		-	-	7.858	-	-	7.858
$Z_t$ (MPa)		9.500			9.500
$S_t$ (MPa)		17.950			17.950
Interlaminar Stress Fraction, $F_{rc-i}$		0.650			0.650
Failure Criteria		0.500			0.510

and

Table 3. Failure criteria the tubular hybrid joint: FEA.

Variable	Value	FEA
		A (0,0,0) mm			B (0,75,0) mm
		$\mathit{\sigma }$	$\mathit{\tau }$		$\mathit{\sigma }$	$\mathit{\tau }$
		(MPa)	(MPa)		(MPa)	(MPa)
		z	yz	xz	z	yz	xz
T (N/mm)	4899	−0.006	-	8.152	−0.006	-	8.152
V (N/mm)	5127	2.330	1.020	-	−0.610	2.480	-
P (N/mm)	134	0.493	−0.393	-	−0.014	0.023	-
${\sigma }_z$ (MPa)		2.817	-	-	-0.629	-	-
${\tau }_{yz}$ (MPa)		-	0.627	-	-	2.503	-
${\tau }_{xz}$ (MPa)		-	-	8.152	-	-	8.152
$Z_t$ (MPa)		9.500			9.500
$S_t$ (MPa)		17.950			17.950
Interlaminar Stress Fraction, $F_{rc-i}$		0.650			0.650
Failure Criteria		0.540			0.520

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 $S_f$ and the design pressure p that the adhesive joint can withstand.

Table 4. Design limit pressure using the failure criteria.

	Pressure Factor	Design Pressure, p (kN/m2)	Failure Criterion
p (initial)	1.000	5.160	0.540
p (maximum)	1.360	7.030	1.000

presents the limit design pressure p, keeping constant interlaminar stress fraction at 0.650. Similarly,

Table 5. Safety factor using the failure criteria.

	Safety Factor	Failure Criterion
Frc-i (initial)	1.540	0.540
Frc-i (minimum)	2.080	1.000

presents as an alternative to the maximum safety factor $S_f$ to which this FRP panel can be subjected, maintaining a constant design pressure of 5.160 kN/m². These indicated limit values of p and S_f application are subject to compliance of the critical layer analysis of FRP panel proposed by SSC-Lloyd’s Register, [39].

The safety factor S_f is defined as:

$$S_f=\frac{1}{F_{rc-i}}$$

(22)

4. 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 8a 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 10 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 Figure 11 and Figure 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 12 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 Figure 6) 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.

5. 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 10⁶ 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.