## 2 Methodology
### Spring Model  
The system is modeled using a set of parallel and serial springs (Figure 11). Pin stiffness (Kp) and shaft stiffness (Ks) were calculated assuming they were cantilever beams under bending while the axial stiffness of delrin (C) under compression was used as stiffness of the bone analogous (KN). The stiffness depends on the length of the component, while the stiffness of the pins is also a function of the force exerted on them (Figure 3). As shaft bending occurs in a tangential plane a conversion factor (t) was defined to convert displacement along the shaft axis. A hypothetical, segmented beam was defined in a length corresponding to a length of a shaft component in the system and was subject to varying bending loads. Loss of length due to bending was calculated against bending distance. No of segments were changed to obtain more data points. Linear regression was used to create a linear relationship between these 2 parameters. Even though the relationship is clearly not a linear relationship this method was used to simplify usage in a spring model where the F = kx form is preferred. This method was replicated for all other lengths of shaft segments found in the system test (45 cm, 90 cm, 135 cm, and 180 cm) (Figure 12). Based on initial calculations it was identified that shaft bending and compression both provided significant input to the overall deformation. Therefore, the shaft spring constant was defined as 2 springs in series with bending and compression. \[K=(K_{N3}.K_{1}.K_{S2}.K_{2}.K_{N4})/(K_{N3}+K_{1}+K_{S2}+K_{2}+K_{N4}) \tag{4}\]  
where: \(K_{1}=(K_{N}1.K_{P}1)/(K_{N}1+K_{P}1)+(K_{S}1.K_{P}3)/(K_{S}1+K_{P}3)\)  
& \(K_{2}=(K_{N}2.K_{P}2)/(K_{N}2+K_{P}2)+(K_{S}3.K_{P}4)/(K_{S}3+K_{P}4)\)  
Figure 10: Simplification of system. System is divided into 2 sections from the mid-point of the fracture. x1-vertical distance from mid-point of fracture site to center of pin 1. x2-vertical distance from mid-point of fracture site to center of pin 2. F1, F2-vertical force acting on each pin. Figure 11: Simplified system. Spring constants of each segment calculated based on their material properties and type of deformation. N-Bone analogous, P-Pin, and S-Shaft. Figure 12: **Top**: Segmented model with each segment considered a stiff shaft with no deformation and 2D simulation. **Bottom**: scatter plot of vertical displacement due to bending(y) and length reduction horizontally (x). Regression lines: Beam length 180 cm (Red), 135 cm (Magenta), 90 cm (Blue), and 45 cm (Black). The pin and clamp behavior observed was used to create a simplified FEA model. The pin clamp assembly was substituted with a simple pin and block to reduce the time and cost of computation. The material properties of the pin were defined to mimic a new material undergoing bi-linear hardening, to replicate the slippage occurring in the pin clamp assembly. Material properties were obtained via calculations using the interface test results and measurements obtained. These material properties were used to recreate the interface test virtually for validation. The obtained material properties were used in a FEA model of the complete system and compared against test results. A comprehensive finite element analysis was conducted using a geometry specific model as a benchmark. The model was generated in Solidworks(tm) using mechanical drawings to obtain necessary geometry (Figure 13). Linear elastic isotropic material properties were applied for the material using literature and test results (Table 2). Boundary conditions provided were based on the testing protocol. Bone-pin interface was provided bonded properties while clamp, shaft, and pin interfaces amongst each other were provided a friction co-efficient of 0.6. Linear static analysis was conducted using the finite element method in the software ANSYSYtm. The mesh sensitivity study was conducted on the geometry specific model to identify the optimum mesh size for the system. Meshing was done in such a manner that meshes in both methods had similar refinement for the ease of comparison.