## 4 Example Analysis of MFC Laminated Beam Structure  
To explain the precision of the MFC actuating force formula based on SSDT, the actuating experiment and finite element analysis of the MFC laminated beam in [29] is exemplified for analysis and discussion. It adopts the M2814-P1 and M2814-P2 types of MFC, with the distance between the raised position and the fixed end being 10 mm. The MFC laminated beam structure is shown in Figure 6, and the parameters of the P1 type and the P2 type of MFC and those of the beam's structure in are shown in Table 1. The MFC actuating force formula based on SSDT deduced in this paper is used to calculate the mean internal force  
Figure 7 shows that under a fixed voltage, the MFC internal force along the direction of the \(y\) axis is much smaller than that along the direction of the \(x\) axis, meaning the direction of the \(x\) axis is the main actuating direction in the P1 type of MFC. It states that the deformability of the P1 type along the direction of the \(x\) axis is better than that along the direction of the \(y\) axis, conforming to its characteristic deformation. In the directions of the \(x\) and \(y\) axes, the marginal internal force of the P1 type at the stickup position is small, the internal force increases rapidly towards the central position and tends to be stable, and the MFC in the direction of the \(x\) axis takes on a compression state while the MFC in the direction of the \(y\) axis takes on a tension state. Similarly, Figure 8 shows the same characteristics for the P2 type. This is created by the shear lag between the MFC and the beam structure, which reflects that at the border position of the MFC, the interfacial shear stress is much greater than that at the center and the shear stress tends to sharply declining from the positions at the border towards the center. This makes the MFC normal stress mutate at border positions, which take on hyperbolic cosine function distributions. According to the MFC's internal force distribution, its shear deformation is one of the primary factors on actuating force transmission. In [29], actuating experiments were performed on a beam structure attached to the P1 and P2 type MFCs, respectively. Displacement and differences compared between the maximum and minimum end displacement were made in two working conditions of the same thickness and different thicknesses under changing voltages, respectively. As for the P1 MFC, the voltages of the maximum and minimum end displacement were \(-500\) V and \(1500\) V; as for the P2 type, the voltages of the maximum and minimum end displacement are \(240\) V and \(-120\) V. In the first place, the MFC actuating formulas based on SSDT and on CPT were used to conduct a simulation calculation on the end displacement of an MFC laminated beam of \(1.5\) mm thickness using ANSYS finite element software. In a simulation calculation of the MFC actuating force based on SSDT, MFC sub-sectional actuating forces were calculated with an effective length of \(0.5\) mm in the actuating direction, and then exerted to the corresponding sticking position through the surface effect unit in ANSYS. When the MFC actuating formula based on CPT was used, the MFC actuating force was directly exerted to the fringe nodes at the stickup position for the simulation calculation. The two simulation results are finally compared with the experimental result in [29], as shown in Figure 9. It can be seen that the simulation result from the MFC actuating force formula based on SSDT is closer to the experimental result. The maximum deviation is \(2.45\%\) between the MFC actuating force simulation results based on SSDT attached to the P1 type of MFC and the experimental results, while the maximum deviation is \(9.07\%\) between the MFC actuating simulation results based on CPT and the experimental results.