# Parameter Identification of Structures with Different Connections Using Static Responses
## 2 Formulation for Parameter Identification  
In the parameter-identification algorithm, the difference between the analytical and measured displacements is defined using the objective function. The unknown parameters can be obtained by determining the global-minimum value of the objective function. In this method, adequate excitation is imparted to the structure using the applied static forces. The damage condition of the components of the structure was determined on the basis of the measurement of nodal displacements of the structure.  
The analytical displacements can be obtained using the stiffness method [25]. This method is based on the force-displacement relationship for the structure, as expressed in Equation (1):  
\[\mathbf{Q}=\mathbf{K}\mathbf{D} \tag{1}\]  
where \(\mathbf{Q}\) and \(\mathbf{D}\) represent the global forces and displacements, respectively; \(\mathbf{K}\) is the entire-structure stiffness matrix that can be obtained by assembling the member stiffness matrix in global coordinates, \(\mathbf{k}\), where \(\mathbf{k}=\mathbf{T}^{T}\ \mathbf{k}^{\prime}\ \mathbf{T}\), where \(\mathbf{k}^{\prime}\) is the member stiffness matrix in local coordinates, and \(\mathbf{T}\) is referred to as displacement transformation matrix. \(\mathbf{K}\) contains \(P\), which is a set of unknown parameters (such as unknown cross-sectional area \(A_{u}\), unknown moment of inertia \(I_{u}\), unknown nodal rotational stiffness \(R_{u}\)) that need to be identified. Equation (1) can be divided to separate the known and unknown displacements as under:  
\[\left[\begin{array}{c}\mathbf{Q}_{k}\\ \mathbf{Q}_{u}\end{array}\right]=\left[\begin{array}{cc}\mathbf{K}_{11}& \mathbf{K}_{12}\\ \mathbf{K}_{21}&\mathbf{K}_{22}\end{array}\right]\left[\begin{array}{c} \mathbf{D}_{u}\\ \mathbf{D}_{k}\end{array}\right] \tag{2}\]where \(\mathbf{Q}_{k}\) and \(\mathbf{D}_{k}\) are the known external loads and boundary conditions of the nodal displacements, respectively. \(\mathbf{Q}_{u}\) and \(\mathbf{D}_{u}\) are the unknown loads and boundary conditions of the nodal displacements, respectively. By solving Equation (2), we obtain:  
\[\mathbf{D}_{u}=\mathbf{K}_{11}^{-1}(\mathbf{Q}_{k}-\mathbf{K}_{12}\mathbf{D}_{ k}) \tag{3}\]  
The objective function is defined as the sum of the square of the difference between the analytical displacement \(\mathrm{D}_{u}\) and measured displacement \(\mathrm{D}_{m}\); \(n\) is the total number of measured displacements.  
\[f=\sum_{i=1}^{n}\left(\mathrm{D}_{u}{}^{i}-\mathrm{D}_{m}{}^{i}\right)^{2} \tag{4}\]  
The unknown parameter \(P\) of the structure can be obtained by minimizing the objective function. Assume that the optimal value of \(P\) is \(P^{*}\), which can be determined using Equation (5):  
\[P^{*}=\underset{P}{\mathrm{argmin}}(f) \tag{5}\]  
Rod structures are composed of elements and joints. The joints are used to transfer load from one structural element to another; the joints include pin, rigid, and semi-rigid connections. The form of member stiffness matrix \(\mathbf{k}^{\prime}\) should be distinguished in the process of forming the damage-identification objective function for truss structure, rigid frame, and semi-rigid frame structures.