## 4 Mechanical Model and Numerical Analysis
### Spring-Mass Model  
The stiffness of the linear elastic is fixed in this model. If plasticity is to be considered, a nonlinear spring with stiffness varying with strain (deformation) must be used. The spring-mass model is usually used to discuss rivet joints [25, 26]. The study indicated that the stress in a riveted structure is primarily determined by the rivet transfer load and the bypass load, while the interference force has a smaller effect [27]. The force of the lap plate is shown in Figure 18 where \(F_{t}\) represents the tensile load, Fri is the rivet transfer load, and Fai is the bypass load. The rivet transfer load can be calculated using Equation (9):  
\[F_{t}=\sum_{i}^{N}F_{r}^{i} \tag{9}\]  
where \(N\) is the number of rivet lines. According to its definition, the bypass load can be expressed by Equation (10):  
\[F_{a}^{i}=F_{a}^{i+1}+F_{r}^{i+1} \tag{10}\]  
The bypass load calculation is based on the rivet transfer load; therefore, calculating the rivet transfer load is crucial for analyzing the load distributions of the riveted members. Equations (9) and (10) show that the algebraic sum of all the rivet transfer loads is equal to the plate tensile load, \(F_{t}\). The bypass load that is transferred backward around the previous rivet hole is equal to the sum of the external loads of all the subsequent rivets; therefore, no load is transferred backward around the last hole. There are significant differences between the bypass loads at different locations, which result in differences in  
Figure 17: Schematic diagram of the stress state around a riveted hole. Figure 18: Multi-rivet receiving diagram. the stress distributions at different locations in riveted lap joints. The bypass load causes the outer-row stress to be consistently larger than the inner-row stress. During a load analysis of a lap joint, the spring-mass model can be used to fit the load transfer process when the member is loaded, as shown in Figure 19. In Figure 19a shows the simplified model, in which \(W\) is the plate width, \(a\) is the row spacing, \(r\) is the rivet radius, \(t_{1}\) and \(t_{2}\) are the thicknesses of the upper and lower plates, respectively, and Figure 19b depicts the spring-mass model, where KA and KC represent the spring stiffnesses of the upper and lower plates, respectively, and KB is the spring stiffness of the rivet. The rivet-hole contact surface was treated as a rigid body with no mass, and the plate and rivet between the contact surfaces were replaced with a spring that had a coefficient of elasticity. Since the tensile direction of the plate and the shear deformation direction of the rivet are both horizontal, the axial deformation can be ignored in this direction. Therefore, the plate and rivet were reduced to a spring with elasticity, but only in the horizontal direction. According to [PERSON]'s law, the load transmitted by the spring is equal to the product of the coefficient of elasticity and the displacement difference between the two ends of the spring. Therefore, Equation (11) can be used for each mass:  
\[\left\{ \begin{array}{c}FA_{i}=KA_{i}(x_{Ai}-x_{Ai+1}-2\triangle_{i})\\ FC_{i}=KC_{i}(x_{Ci}-x_{Ci+1}-2\triangle_{i})\\ FB_{i}=KB_{i}(x_{Ci}-x_{Ai})\end{array} \right., \tag{11}\]  
where \(F_{Ai}\), \(F_{Ci}\), and \(F_{Bi}\) are the loads transmitted by each spring, \(K_{Ai}\), \(K_{Ci}\), and \(K_{Bi}\) are the spring stiffnesses of the upper plate, the lower plate, and the rivet, respectively, \(x_{Ai}\) and \(x_{Ci}\) are the displacements of each mass, and \(\Delta\) is the amount of interference on one side. The equivalent elasticity coefficient of the plate can be calculated using Equation (12):  
\[\left\{ \begin{array}{c}KA=\frac{E_{b}\times W\times t_{1}}{a-2r}\\ KC=\frac{E_{b}\times W\times t_{2}}{a-2r}\end{array} \right., \tag{12}\]  
where \(E_{b}\) is the modulus of elasticity of the plate, \(W\) is the plate width, \(a\) is the row spacing, \(r\) is the radius of the rivet, and \(t_{1}\) and \(t_{2}\) represent the thicknesses of the upper and lower plates, respectively.