# Stress Dispersion Design in Continuum Compliant Structure toward Multi-DOF Endoluminal Forceps
## 2 Stress Dispersion Design Method
### Design Improvement  
A beam of uniform strength is a beam whose bending stress is constant in the longitudinal direction of the beam. For example, when considering a cantilever beam that receives a concentrated load on the tip, the bending moment acting on the beam increases proportionally to the distance from the tip. Therefore, if the cross-section is uniform, the magnitude of bending stress is also proportional to the distance tip and becomes maximum at the fixed end. If a beam is designed so that the section modulus increases proportionally to the distance from the tip, the bending stress becomes constant regardless of the position. In contrast, no straightforward method exists to determine the hinge size to distribute the stress effectively because the designed forceps relies on a discontinuous beam. In this study, we propose a method to effectively determine the parameters by using the beam of uniform strength concept. That is, assume a concentrated load \(P\) acting on the free end of the cantilever shown in Figure 2, where \(x\) is the arbitrary position in the length direction. The cross-sectional shape is a rectangle with a width of \(b\) and a thickness of \(h_{x}\) at an arbitrary position in the length direction. \(h_{0}\) is the thickness at \(x=l\). When \(b\) is constant, \(h_{x}\) becomes parabolic [51]. The applied moment \(M\) and the section modulus \(Z\) are expressed as follows:  
\[M=Px,Z=\frac{{bh_{x}}^{2}}{6} \tag{1}\]  
Thus, the maximum bending stress \(\sigma_{max}\) is expressed as follows:  
\[\sigma_{x(max)}=\frac{M}{Z}=\frac{6 Px}{{bh_{x}}^{2}} \tag{2}\]  
\[\sigma_{l(max)}=\frac{6 Pl}{{bh_{0}}^{2}} \tag{3}\] \begin{table}
\begin{tabular}{p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt}} \hline \hline  & & & & & & & \\ \cline{3-8}  & & & & & & & \\ \cline{3-8}  & & & & & & & \\ \hline Classification & Discrete joint & & & & & & \\ \cline{3-8}  & & & & & & & \\ \cline{3-8}  & & & & & & & \\ \hline Parts of joint & Discrete & Compliant & Compliant & Compliant & Compliant & Compliant \\ \hline Symmetry & Symmetric & Asymmetric & Asymmetric & Asymmetric & Asymmetric & Asymmetric & Symmetric \\ \hline Direction of bending & & & & & & & \\ \cline{3-8}  & & & & & & \\ \cline{3-8}  & & & & & & \\ \hline DOF of bending & 1-DOF & 1-DOF & 1-DOF & 1-DOF & 2-DOF & 2-DOF \\ \hline Advantage & & & & & & & \\ \cline{3-8}  & & & & & & \\ \cline{3-8}  & & & & & & \\ \hline Disadvantage & & & & & & & \\ \cline{3-8}  & & & & & & \\ \cline{3-8}  & & & & & & \\ \hline Reference & [29; 30; 31] & [32; 33; 34; 35] & [36; 37; 38; 39; 40; 41] & [42; 43] & [44; 45; 46] & [47; 48; 49; 50] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Classification of several hinge types in related studies: (i) discrete joint, (ii) uni-directional compliant joint, (iii) bi-directional compliant joint, (iv) cross-axis compliant joint, (v) helical compliant joint, and (vi) symmetric compliant joint. where \(\sigma_{x(max)}\) and \(\sigma_{l(max)}\) are the maximum bending stress at an arbitrary position in the length direction and the base end, respectively. Considering a beam of uniform strength, \(\sigma_{x(max)}=\sigma_{l(max)}\). Therefore,  
\[\sigma_{x(max)}=\frac{6 Px}{bh_{x}{}^{2}}=\frac{6 Pl}{bh_{0}{}^{2}} \tag{4}\]  
\[h_{x}=h_{0}\sqrt{\frac{x}{l}} \tag{5}\]  
Due to the size limitations of the endoluminal forceps, \(h_{0}=0.6\) mm, \(l=30\) mm. By substituting these parameters into Equation (5), the thickness of the beam of uniform strength can be calculated. The hinge thickness was determined according to the thickness for each length of the beam of uniform strength. In the proposed method, average thicknesses of two pairs of adjacent hinges are matched with thickness of beam of uniform strength. The thicknesses for each position in the length direction of beam of uniform strength, initial design, and improved design are shown in Figure 3. The hinge lengths were designed as (a) 0.2 mm, (b) 0.4 mm, (c) 0.8 mm, and (d) 1.2 mm.