# Observing Material Properties in Composite Structures from Actual Rotations
## 3 Results
### Example 1: Simply Supported Beam with Two-Iteration Processes  
Consider the 3-m long and 0.2-m wide simply supported beam modeled with three nodes and two beam elements depicted in Figure 4a. A deformed shape with the value of bending vertical deformation and with bending and shear vertical deformation at node 2 can be seen in Figure 4b and a deformed shape with bending rotation and with bending and shear rotation at node 1 is illustrated in Figure 4c. This beam has a rectangular constant cross-section with a 1m\({}^{2}\) cross-sectional area. Its mechanical properties are listed in Table 1. The boundary conditions of the structure are horizontal and vertical displacements restricted in node 3 and vertical displacement is restricted in node 1 (this is to say v\({}_{1}\) = u\({}_{3}\) = v\({}_{3}\) = 0). The beam is subjected to a concentrated vertical force in node 2 of 100 kN (V\({}_{2}\) = 100 kN). For the inverse analysis of the structure, the load V\({}_{2}\), the length of the elements, Poisson's ratio, and Young's modulus E are assumed as known, while the inertia I, and the shear area A\({}_{\rm v}\) for both elements are assumed as unknown. To identify the two unknown parameters (A\({}_{\rm v}\) and I) a measurement set of at least four rotations is required to identify the values of the unknowns. As explained in Section 2, the COM is based on the optimization application, therefore, all variables are always observed. Regardless of their accuracy. In order to ensure the reliability of the COM-based model, it is advised to perform a traditional OM which is based on the linear equation. The analysis of the traditional OM of this model proves that no set of four rotations in the OM enables the proper identification of the unknown parameters. Therefore, a set of vertical deflection and a rotation should \begin{table}
\begin{tabular}{c c} \hline
**Properties** & **Value** \\ \hline axial stiffness [GPa-m\({}^{2}\)] & 440.65 \\ shear stiffness for node 1 [GPa-m\({}^{2}\)] & 344.05 \\ shear stiffness for node 2 [GPa-m\({}^{2}\)] & 385 \\ flexural stiffness for node 1 [GPa-m\({}^{4}\)] & 1246.7 \\ flexural stiffness for node 2 [GPa-m\({}^{4}\)] & 1120 \\ \hline \end{tabular}
\end{table} 
Table 1: Properties of the FEM of the simply supported beam. Figure 3: Flowchart of 2 COM. be measured together (it is to say w1, w2, w3 and v2). The values of the measurement set obtained from [PERSON]'s beam theory simulation can be seen in Figure 4b,c. To check the robustness of the method, a set of random values is chosen for the inertia and shear area to be used to calculate the hypothetical site measurements. A total of 50 different random sets of the initial value coefficients for the inertia and the shear area were randomly chosen between 0.8 and 1.2 of the theoretical values. These initial parameters are successively updated throughout the iterative process. From 50 analyses, in three analyses the optimization process was not convergent to the solution and no result was acquired. The normalized average, standard deviation, and coefficient of variation of the remaining results as well as the mode are presented in Table 2.  
These results prove the adequate convergence of the proposed methodology in simply supported beams for random initial values of the inertia. The reason for the flaw in the observed value of GAv2 is that the measurement on node 2 cannot provide sufficient \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Average** & **CoV** & **Standard Deviation** \\ \hline EI1 & 1.000 & 8.791 \(\times\) 10\({}^{-7}\) & 8.791 \(\times\) 10\({}^{-7}\) \\ EI2 & 1.000 & 9.114 \(\times\) 10\({}^{-7}\) & 9.114 \(\times\) 10\({}^{-7}\) \\ GAv1 & 1.000 & 0.000 & 0.000 \\ GAv2 & 0.910 & 0.128 & 0.116 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Obtained results from random initial values of example 1. Figure 4: Example 1. (**a**) FEM for a simply supported beam. (**b**) Deformed shape with the value of bending vertical deformation and with bending and shear vertical deformation at Node 2.