## 2 Modeling and Governing Stress Equations
\[\frac{du_{0i}}{dx}=\frac{B_{11}^{i}}{\left(B_{11}^{i}\right)^{2}-D_{11}^{i}A_ {11}^{i}}M_{i}+\frac{D_{11}^{i}}{\left(B_{11}^{i}\right)^{2}-A_{11}^{i}D_{11}^ {i}}N_{i} \tag{14}\]  
\[\frac{d^{2}w_{i}}{dx^{2}}=\frac{A_{11}^{i}}{\left(B_{11}^{i}\right)^{2}-A_{11 }^{i}D_{11}^{i}}M_{i}+\frac{D_{11}^{i}}{\left(B_{11}^{1}\right)^{2}-A_{11}^{i} D_{11}^{i}}N_{i} \tag{15}\]  
In these equations, the upper indices of \(A_{11}^{i}\), \(B_{11}^{i}\) and \(D_{11}^{i}\) (elements of \(A\), \(B\), and \(D\) matrices) depict the number of the adherends (\(i\) = 1-3). On the other hand, according to FBD (Figure 4), there are relationships between forces and moments (around point \(A\)) of three adherends, as shown in Equations (16) and (17). \[N_{2}=2T-N_{1}-N_{3} \tag{16}\]  
\[M_{2}=2T\left(\frac{t_{2}}{2}+t_{a}+\frac{t_{3}}{2}\right)-M_{1}-M_{3}-N_{1} \left(\frac{t_{1}}{2}+t_{2}+2t_{a}+\frac{t_{3}}{2}\right)-N_{2}\left(\frac{t_ {2}}{2}+t_{a}+\frac{t_{3}}{2}\right) \tag{17}\]  
Combining Equations (14) and (15) with the first derivative of Equation (11) and substituting Equations (16) and (17) into the resulting expression while using the first derivative of Equation (2), we have  
\[\frac{d^{2}N_{1}}{dx^{2}}=-\frac{G_{a}}{t_{a}}(k_{1}M_{1}+k_{2}N_{1}+k_{3}M_{3 }+k_{4}N_{3}+k_{5}T) \tag{18}\]  
Figure 4: Free body diagram of the left hand side of the joint. Additionally, by implementing Equations (15)-(17) in the first derivative of Equation (12),  
\[\frac{d^{2}\sigma_{1}}{dx^{2}}=\frac{E_{a}}{t_{a}}\left(k_{6}M_{1}+k_{7}N_{1}+k_{8 }M_{3}+k_{9}N_{3}+k_{10}T\right) \tag{19}\]  
where _ki_ (_i_ = 1-10) are constraints of thicknesses of the joint and stiffness matrix arrays as expressed in Appendix A. In the same way, it is possible to find similar equations for \(N_{2}\) and \(\sigma_{2}\). Equation (20) is formed by derivation and a combination of Equations (8)-(10). \[\frac{d^{4}M_{1}}{dx^{4}}=\frac{d^{2}\sigma_{1}}{dx^{2}}-\frac{d^{4}N_{1}}{dx^ {4}}\left(\frac{t_{a}+t_{1}}{2}\right) \tag{20}\]  
Now, substitute Equation (19) into Equation (20). \[\frac{d^{4}M_{1}}{dx^{4}}=k_{6}M_{1}+k_{7}N_{1}+k_{8}M_{3}+k_{9}N_{3}+k_{10}T- \frac{d^{4}N_{1}}{dx^{4}}\left(\frac{t_{a}+t_{1}}{2}\right) \tag{21}\]  
Due to the symmetry of the joint, these relationships are correct: \(N_{1}=N_{3}\) and \(M_{1}=M_{3}\). To substitute them in equations obtained for forces and moments and also in Equations (16) and (17), the differential equation of changes according to \(x\) is as follows:  
\[\frac{d^{6}N_{x1}}{dx^{6}}+\delta_{1}\frac{d^{4}N_{x1}}{dx^{4}}+\delta_{2} \frac{d^{2}N_{x1}}{dx^{2}}+\delta_{3}N_{x1}=\delta_{4}T \tag{22}\]  
where coefficients \(\delta_{i}(i=1-4)\) are calculated via equations of \(k_{i}\) as mentioned in Appendix A. So, the changes in horizontal force \(N_{1}\) are calculable by solving differential Equation (22):  
\[\frac{d^{2}w_{i}}{dx^{2}}=\frac{A_{11}^{i}}{\left(B_{11}^{i}\right)^{2}-A_{11 }^{i}D_{11}^{i}}M_{i}+\frac{D_{11}^{i}}{\left(B_{11}^{1}\right)^{2}-A_{11}^{i} D_{11}^{i}}N_{i} \tag{15}\]0  
where \(c_{i}\) (_i_ = 1-6) are differential equation coefficients that boundary conditions can obtain. Additionally, \(q_{i}\) (_i_ = 1-3) are parameters related to \(\delta_{i}\) that are expressed in Appendix A. Now, it is possible to find the peel stress equation from Equation (23) that expresses the force variations along the \(x\)-axis. At first, it is necessary to find the trend of bending moment changes along the \(x\)-axis. So the \(M_{1}\) equation is formed from the symmetry assumption, and Equation (21) is as follows:  
\[ \begin{array}{c}M_{1}(x)=\theta_{1}c_{1}exp^{q_{1}x}+\theta_{1}c_{2}exp^{-q_ {1}x}+\theta_{2}c_{3}exp^{x\sqrt{q_{2}+q_{3}i}}+\theta_{3}c_{4}exp^{x\sqrt{q_ {2}-q_{3}i}}+\theta_{2}c_{5}exp^{-x\sqrt{q_{2}+q_{3}i}}\\ \hskip 14.226378 pt+\theta_{3}c_{6}exp^{-x\sqrt{q_{2}-q_{3}i}}-\frac{Tk_{5}}{k _{1}+k_{3}}\end{array}  \tag{24}\]  
where \(\theta_{i}\) (_i_ = 1-3) are calculated via equations explained in Appendix A. Additionally, according to Equation (8),  
\[