## 2 Methodology
### Proposed Formulas for Tubular Hybrid Joint
#### 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].  
\[\frac{d^{4}\sigma_{z}}{\partial y^{4}}+4K_{5}{}^{4}\sigma_{z}=0, \tag{1}\]  
where:  
\[K_{5}{}^{4}=\frac{E_{a}}{4\cdot t}\bigg{(}\frac{1}{D_{1}}+\frac{1}{D_{2}}\bigg{)}, \tag{2}\]using the following compliance factors for the normal stress calculation:  
\[\beta_{1}=\frac{12\cdot E_{a}\cdot\left(1-{\mu_{1}}^{2}\right)}{E_{1}\cdot h_{1} ^{3}\cdot t}, \tag{3}\]  
\[\beta_{2}=\frac{12\cdot E_{a}\cdot\left(1-{\mu_{2}}^{2}\right)}{E_{2}\cdot h_{2 }^{3}\cdot t}. \tag{4}\]  
Based on compliance of these two factors (Equations (3) and (4)), the following formulas are obtained to determine the maximum values of normal stresses:  
\[\sigma_{V_{c}}=\frac{-(2)^{0.5}\cdot\beta_{1}\cdot V_{c}}{(\beta_{1}+\beta_{2} )^{0.75}}, \tag{5}\]  
\[\sigma_{M_{c}}=\frac{-\beta_{1}\cdot M_{c}}{(\beta_{1}+\beta_{2})^{0.5}}. \tag{6}\]  
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.  
\[\frac{d^{3}\tau_{yz}}{\partial y^{3}}-K_{6}^{2}\frac{d\tau_{yz}}{\partial y}=0, \tag{7}\]  
where:  
\[{K_{6}}^{2}=\frac{4\cdot G_{a}}{t}\Bigg{(}\frac{\left(1-{\mu_{1}}^{2}\right) }{E_{1}\cdot h_{1}}+\frac{\left(1-{\mu_{2}}^{2}\right)}{E_{2}\cdot h_{2}} \Bigg{)}, \tag{8}\]  
using the following compliance factors for the calculation of shear stress:  
\[\alpha_{1}=\frac{G_{a}\cdot\left(1-{\mu_{1}}^{2}\right)}{E_{1}\cdot h_{1} \cdot t}, \tag{9}\]  
\[\alpha_{2}=\frac{G_{a}\cdot\left(1-{\mu_{2}}^{2}\right)}{E_{2}\cdot h_{2} \cdot t}. \tag{10}\]  
Figure 2: Typical diagram of the stiffener-panel joint.  
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:  
\[\tau_{T_{c}}=\frac{-\alpha_{1}\cdot T_{c}}{2\cdot(\alpha_{1}+\alpha_{2})^{0.5}}, \tag{11}\]  
\[\tau_{V_{c}}=\frac{3\cdot V_{c}}{4\cdot h_{1}}, \tag{12}\]  
\[\tau_{M_{c}}=\frac{3\cdot\alpha_{1}\cdot M_{c}}{h_{1}\cdot(\alpha_{1}+\alpha_{2 })^{0.5}}. \tag{13}\]  
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.