## 3 Model Validation
### Experimental Study of Large Geometric Nonlinear Deformation of Flexible Beams  
The numerical examples in this section will explore the static behavior of beams. The example illustrates the highly nonlinear behavior displayed by a cantilever beam when subjected to end loading. Through this deformation example, the effects of material model settings and stress assumption conditions are demonstrated. Assuming the beam remains in the linear elastic stage, its parameters are detailed in Table 1, and the structure of the cantilever beam is depicted in Figure 5.  
During deformation, the centerline of the cantilever beam undergoes a rotation, akin to assumptions made in mechanical analysis. Therefore, the longitudinal position vector gradient is designated as a free variable. In addition to the associated slope, both the translational nodal displacements and the slope vectors at the beam's constrained ends  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Definition** & **Numerical** & **Unit** \\ \hline Initial Length & 2.0 & m \\ Initial Width & 0.04 & m \\ Initial Height & 0.04 & m \\ Section Modulus & 5/6 & / \\ Second Moment of Area & \(8.533\times 10^{-5}\) & m\({}^{6}\) \\ Mass Density & 5540 & kg/m\({}^{3}\) \\ Poisson’s Ratio & 0.3 & / \\ Young’s Modulus & \(2.01\times 10^{7}\) & Pa \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of the beam with tip load.  
Figure 5: Concentrated load acting on the cantilever beam.  
are restricted. Consequently, the slope of the centerline assumes a non-zero angle with the horizontal direction, aligning with the Timoshenko beam theory.  
Particular attention was devoted to the influence of shear effects in displacement analysis. For short and slender beams, the end load was configured at \(F=1\times 10^{6}h^{3}\) (Unit: N), and the obtained displacement results were subsequently compared with those from the analysis conducted under the same conditions using ANSYS software (2024 R1). In ANSYS, the model utilized the Beam188 element type. The deformation of the beam is depicted in Figure 6, and the analysis results in Table 2 detail the end positions of the beam using different numbers of elements.  
Table 2 demonstrates that the deformation of the cantilever beam under the action of end loading using ANCF elements closely aligns with the deformation calculated by ANSYS, with the error in deformation angles remaining within a very small range. This result suggests that the model proposed in this paper provides reasonable and accurate elastic deformation when subjected to predetermined stresses. Furthermore, comparisons of the results of cantilever beams with different numbers of elements to analytical data reveal that all models exhibit convergent behavior. This further underscores the reliability of this method.  
For long and slender beams, a slenderness ratio of \(S_{L}\) is used to describe the significance of shear deformation in the beam. In this example, the end load of the beam is set to  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**End Position and Deformation Angle**} \\ \cline{2-4}
**Number of Elements** & \multicolumn{3}{c}{**End Position (x, y)**} & **Deformation Angle** \\ \hline
2 & 1.989005 & \(-\)0.198323 & \(-\)0.099381 \\
3 & 1.988033 & \(-\)0.198376 & \(-\)0.099456 \\
4 & 1.987799 & \(-\)0.198350 & \(-\)0.099455 \\
5 & 1.987796 & \(-\)0.198354 & \(-\)0.099456 \\
6 & 1.987798 & \(-\)0.198369 & \(-\)0.099464 \\ ANSYS & 1.987795 & \(-\)0.198361 & \(-\)0.099460 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Deformation position of the short and slender cantilever beam, \(S_{\text{L}}=50\).  
Figure 6: Deformation diagram of cantilever beam under end concentrated load.  
\(F=1\times 10^{4}h^{3}\)(Unit: N), the initial length is adjusted to 20 m, and the slenderness ratio is \(S_{L}=500\), while other conditions remain unchanged. As shown in Table 3, the deformations predicted by the ANCF elements appear to be slightly larger than those obtained using ANSYS.