# An Improved Winkler Foundation Modulus for a Beam in a Full Space
## 3 Verification  
### Numerical Verification  
In this subsection, the solutionsfrom the beam-spring model using the proposed Winker foundation modulus and the Winker foundation modulus proposed by [PERSON] et al. [20] are compared with three-dimensional solid finite element solutions obtained from ABAQUS.  
As shown in Figure 6, a three-dimensional solid finite element model is built. The model has a length of 160 m, a width of 80 m, and a height of 80 m. Meanwhile, the beam has an outer diameter of 6.2 m and an inner diameter of 5.5 m. In the middle of the model, the model grid is set as 1 m along the \(x\)-axis. The other parts are sparsely meshed. The same mesh method is applied to the section grid of the model. The ground has a shear modulus of 50 Mpa and a Poisson's ratio of 0.38. The elastic modulus and Poisson's ratio of the beam are 34.5 Gpa and 0.2, respectively. The contact state at the beam-ground interface is a'slip' state. The free-field displacement has the form \(u_{g}(x)=0.1e^{-\frac{1}{2}x^{2}}\), which was obtained from the research of [PERSON] et al. [23] and is applied to the boundaries of the model.  
The beam-spring numerical model obtained from ABAQUS is plotted in Figure 7, wherein the beam is modeled using a Euler beam, which rests on a series of springs. The beam is meshed at 0.1 m. The values of the spring are calculated using the proposed formula for the Winkler foundation modulus and that by [PERSON] et al. [20]. The free-field displacement is input at the far end of the spring. The material parameters of the beam are the same as those used in the three-dimensional solid finite element model.  
\begin{table}
\begin{tabular}{c c c} \hline \(m\) & \(\overline{x}_{0}\) & \(b\xi\) \\ \hline
0.01 & 35.72 & 0.032 \\
0.025 & 16.73 & 0.067 \\
0.04 & 11.43 & 0.098 \\
0.05 & 9.39 & 0.119 \\
0.07 & 6.94 & 0.161 \\
0.1 & 5.06 & 0.220 \\
0.12 & 4.4 & 0.253 \\
0.2 & 2.81 & 0.396 \\
0.25 & 2.27 & 0.490 \\
1 & 0.57 & 1.949 \\ \hline \end{tabular}
\end{table}
Table 1: Fitting data from the two methods.  
For further explanation, a relative error is defined as \((Ce-Ca)/Ce\), in which \(Ce\) is the peak response of the beam from the numerical model, as well as the peak response \(Ca\) obtained from the model of an Euler beam on the Winkler modulus.  
Figure 8 illustrates the displacement distribution along the beam axis. It is evident that compared with numerical solutions obtained using the solid finite element model, the displacement obtained using the proposed Winkler foundation modulus performs better than those from the Winkler foundation modulus of [PERSON]. To be specific, the beam's displacements using the proposed Winkler foundation modulus and Winkler foundation modulus by [PERSON] have errors of 6.15% and 12.29%, respectively. Figure 9 shows the shear force and bending moment response of the beam obtained using different methods. Evidently, compared with the results obtained using the Winkler foundation modulus by [PERSON], the results adopting the proposed Winkler foundation modulus are closer to the numerical solutions, as for both bending moment and shear force. In detail, compared with the numerical solution, the errors for solutions using the Winkler foundation modulus obtained by [PERSON] are 10.81% for the bending moment and 23.6% for the shear force, respectively. However, the corresponding errors using the proposed Winkler foundation modulus are 2.25% and 7.20%, respectively, which are quite small.  
Figure 6: Three-dimensional solid finite element model.  
Figure 7: Beam–spring numerical model.