# Deflection Modeling and Curvature Manipulation of a Variable-Stiffness Flexible Needle
## 2 Materials and Methods
### Model Analysis of Flexible Needle
The bending moment at position \(x(0\leq x\leq L)\) from the right side can be denoted as  
\[M=Fx,\quad 0\leq x\leq L \tag{5}\]  
Using [PERSON]'s integral method, the expression for the bending moment is  
\[w_{b}=\int_{0}^{L}\frac{M(x)M(x)}{EI(x)}dx,\quad 0\leq x\leq L \tag{6}\]  
where \(E\) is the [PERSON]'s modulus and \(M(x)\) is the bending moment under a virtual unit load (identical to \(M(x)\), i.e., \(M(x)=x\)). In the right segment, there is  
\[\int_{0}^{L_{s}}\frac{Fx\cdot x}{EI_{s}}dx=\frac{FI_{s}^{3}}{3 EI_{s}},\quad 0 \leq x<L_{s} \tag{7}\]  
In the left segment, there is  
\[\int_{L_{s}}^{L}\frac{Fx\cdot x}{EI_{t}}dx=\frac{F(L^{3}-L_{s}^{3})}{3 EI_{t}},\quad L_{s}<x\leq L \tag{8}\]  
Then, the total bending deflection (at the position \(x=0\)) can be denoted as  
\[I_{s} =\frac{\pi R_{s}^{4}}{4} \tag{1}\]0  
In Timoshenko theory, the shear deflection is calculated as  
\[I_{s} =\frac{\pi R_{s}^{4}}{4} \tag{1}\]1  
where \(V(x)=F\) is the actual shear force, \(V(x)=1\) is the virtual unit shear force, and \(A(x)\) is the cross-sectional area. In the right segment, there is  
\[I_{s} =\frac{\pi R_{s}^{4}}{4} \tag{1}\]2  
In the left segment, there is  
\[I_{s} =\frac{\pi R_{s}^{4}}{4} \tag{1}\]3  
\[I_{s} =\frac{\pi R_{s}^{4}}{4} \tag{1}\]4The total deflection at the position \(x=0\) is the sum of the bending and shear contributions:  
\[ \begin{split} w(0)&=w_{b}+w_{s}\\ &=\frac{FL^{3}}{3 EI_{t}}+\frac{FL_{s}^{3}}{3E}\left(\frac{1}{l_{s}} -\frac{1}{l_{t}}\right)+\frac{\kappa F}{G}\left(\frac{L_{s}}{A_{s}}+\frac{L_{t }}{A_{t}}\right)\end{split}  \tag{14}\]  
To identify the primary factors influencing deflection, the bending deflection, shear deflection, total deflection, and ratio of bending deflection to total deflection at different stylet extension lengths were calculated. To ensure consistency with subsequent simulations, the following parameters were used: \(F=0.3\cos(pi/6)N,\
u=0.3\), \(E=75,000\) MPa, \(G=2(1+\
u)/E,\kappa=6(1+\
u)/(7+6\
u),R_{s}=0.2\) mm, \(R_{t}~{}=~{}0.4\) mm, \(L~{}=~{}50\) mm. The results are shown in Table 1.  
As shown in Table 1, the bending deflection dominates the total deflection, accounting for over 99.98%. For slender beams, such as flexible needles, shear deformation can be safely neglected. Therefore, the deflection of the flexible needle in Equation (14) can be simplified to  
\[w(0)=\frac{FL^{3}}{3 EI_{t}}+\frac{FL_{s}^{3}}{3E}\left(\frac{1}{l_{s}}-\frac{1} {l_{t}}\right) \tag{15}\]  
where \(FL^{3}/3 EI_{t}\) represents the deflection of a uniform beam under an external force \(F\), i.e., the deflection of the conventional flexible needle with a beveeled tip. The term \((FL_{s}^{3}/3E)\cdot(1/l_{s}-1/l_{t})\) shows the effect of the variable stiffness structure on the deflection. Since the radius of the stylet is smaller than that of the tube, \(I_{s}<l_{t}\), we have \((FL_{s}^{3}/3E)\cdot(1/l_{s}-1/l_{t})>0\), which indicates that the needle deflection increases. The smaller the moment of inertia of the stylet is, the easier it is for the flexible needle to bend; the longer the stylet extends out of the tube, the more significant the bending of the flexible needle becomes. Since the diameters of the stylet and the tube are determined during the needle manufacturing process, the control strategy for the variable-stiffness flexible needle could be as follows: by controlling the length of the stylet protruding from the tube, the stiffness of the needle tip is adjusted, which in turn changes the bending curvature of the needle.