# Observing Material Properties in Composite Structures from Actual Rotations
## 2 Materials and Methods  
For the SMM, the equations of nodal equilibrium might be written as:  
\[[\text{K}]\cdot\left\{\delta\right\}=\left\{\text{f}\right\}, \tag{1}\]  
where the horizontal u, vertical v, and rotational displacements w, are parts of \(\left\{\delta\right\}\), the stiffness matrix [K] covers information about axial stiffness EA, flexural stiffness EI, and the length of the element L. The horizontal forces, vertical forces, and moments are included in the external force vector [f]. The main disadvantage of this method is that the complexity in the system of equations might lead to the following problems: (1) They are not able to include the effect of shear rotation into the SMM methods, so these methods are considered as common [PERSON]'s beam theory SMM (e.g., [23]). (2) Methods which consider the effects of shear rotation due to some assumptions, are not applicable in general complex structures [44]. In the OM, as in any inverse tool, some parameters of the nodal displacements [8] are measured on-site to identify the unknown mechanical properties in the SMM when the geometry, the boundary conditions, and the load case in a certain static load test are determined. To do so, Equation (1) can be rewritten as presented in Equation (2), where all the known quantities are collected into the coefficient matrix [B] and the vector [D]. Therefore, [B] and [D] are known, while all the unknowns are placed into the vector [Z]. Equation (2) can be solved to observe unknown equations with the help of algebraic operations. For more information, readers are recommended to read [42]. \[[\text{B}]\cdot\left\{z\right\}=\left\{\text{D}\right\}, \tag{2}\]  
In the OM, the system is assumed to be linear, and coupled products of variables are considered single variables (such as Young's modulus E, and inertia I are considered as a unique variable EI). Here, N\({}_{\text{A}}\) is the unknown axial stiffnesses and N\({}_{\text{F}}\) is the unknown flexural stiffnesses. Theoretically, the number of measurements should be at least equal to the number of unknowns; therefore, N\({}_{\text{A}}\) + N\({}_{\text{F}}\) measurements are expected to be enough to observe all unknown parameters. [PERSON] et al. [42] found that due to the lack of nonlinear constraints among product variables in OM, in most cases, it is not able to observe the material properties with the minimum required measurements. [PERSON] et al. [42] proposed a COM which solves the system of equations numerically after including the nonlinear constraints. This is to say that the COM does not provide any symbolic solution, as they are only based on the numerical solution of the system. In order to apply the optimization process, the objective function of the COM can be defined as follows:  
\[\left\{\varepsilon\right\}=[B]\cdot\left\{Z\right\}-\left\{D\right\}, \tag{3}\]  
where \(\left\{\varepsilon\right\}\) is the residual of the equations which is a vector with the same number of rows as the original vector [Z]. The objective function of the optimization process is to minimize the square sum of the residuals, \(\varepsilon\) in Equation (3). MATLAB and Optimization Toolbox Release [45] are used to obtain the optimal solution of the objective function. Before starting the optimization process, the objective function should be normalized. The algorithm for SSI by COM is summarized as follows:  
**Step 1:** Apply SSI by OM to check whether any variable is observed. If all unknown parameters are observed, there is no need to go to the COM process, otherwise, go to step 2. **Step 2:** Obtain Equation (2) from the OM and generate the objective function. **Step 3:** Obtain the normalized unknown parameters. **Step 4:** Guess the unknown parameters' initial values, apply bounds for the solution and solve the optimization process, in order to find the least acceptable value for vector \(\left\{\varepsilon\right\}\). A summary of the procedure is shown in the flow chart in Figure 2. For more information about the COM, the reader is directed to [42].