# A Statistical Model Updating Method of Beam Structures with Random Parameters under Static Load
## 2 New Statistical Model Updating Method
### Statistical Model Updating Equation  
Without losing the generality, it is assumed here that the mass of the structural simulation model is invariable compared with the actual structure [27], and the discrepancy between the simulation model and the actual beam structure is mainly due to the variation of the structural stiffness, which is represented as:  
\[\Delta\mathbf{K}=\sum_{i=1}^{n}\alpha_{i}\mathbf{K}_{i} \tag{4}\]  
where \(n\) is the number of the structural elements; \(\alpha_{i}\) is the element updated factor of the \(i\)th element of the beam structure, which is the variation ratio of a structural parameter such as the bending rigidity of the beam; \(\mathbf{K}_{i}\) is an \(N\times N\) dimensional expanded matrix of the \(i\)th element stiffness matrix, where all elements in expanded parts are zero. In regard to the stiffness matrix of the initial simulation model, the updated stiffness matrix, \(\mathbf{K}_{m}\) can be expressed as follows:  
\[\mathbf{K}_{m}=\mathbf{K}_{a}+\sum_{i=1}^{n}\alpha_{i}\mathbf{K}_{i} \tag{5}\]  
Similar to the initial simulation model, the updated stiffness matrix can be condensed as:  
\[\mathbf{K}_{mt}=\mathbf{K}_{mtt}-\mathbf{K}_{mt\theta}\mathbf{K}_{m\theta t}^ {-1}\mathbf{K}_{m\theta t} \tag{6}\]  
where \(\mathbf{K}_{mtt}\), \(\mathbf{K}_{mt\theta}\), \(\mathbf{K}_{m\theta t}\) and \(\mathbf{K}_{m\theta\theta}\) are sub-matrices of \(\mathbf{K}_{m}\), which are related to the translational and rotational degrees of freedom of the updated finite element model respectively like the situation at the initial simulation state. Equations (5) and (6) can be rewritten as:  
\[\mathbf{K}_{mt}=(\mathbf{K}_{att}+\sum_{i=1}^{n}\alpha_{i}\mathbf{K}_{it})-( \mathbf{K}_{at\theta}+\sum_{i=1}^{n}\alpha_{i}\mathbf{K}_{it\theta})(\mathbf{K }_{a\theta\theta}+\sum_{i=1}^{n}\alpha_{i}\mathbf{K}_{i\theta\theta})^{-1}( \mathbf{K}_{a\theta t}+\sum_{i=1}^{n}\alpha_{i}\mathbf{K}_{i\theta t}) \tag{7}\]  
where \(\mathbf{K}_{itt}\), \(\mathbf{K}_{it\theta}\), \(\mathbf{K}_{i\theta t}\) and \(\mathbf{K}_{i\theta\theta}\) are sub-matrices of the expanded matrix \(\mathbf{K}_{i}\), which corresponds to the sub-matrices of the updated stiffness matrix. Taking the first-order partial derivatives of \(\mathbf{K}_{mt}\) with respect to \(\alpha_{i}\), there is:  
\[\left.\frac{\partial\mathbf{K}_{mt}}{\partial\alpha_{i}}\right|_{\alpha_{i}=0}= \mathbf{K}_{lit}-\mathbf{K}_{lit}\mathbf{K}_{d\theta 0}^{-1}\mathbf{K}_{d\theta t}- \mathbf{K}_{at\theta}\mathbf{K}_{a\theta 0}^{-1}\mathbf{K}_{i\theta t}+\mathbf{K}_{at\theta} \mathbf{K}_{a\theta 0}^{-1}\mathbf{K}_{i\theta t}\mathbf{K}_{a\theta t} \tag{8}\]  
It is worth pointing out that in the sensitive methods as presented by [PERSON] et al. in the literature [28; 29], the calculation steps similar to the above are used to obtain the partial derivatives of the mass matrix and stiffness matrix with respect to updating parameters based on the first-order Taylor expansion. Since the applied load is the same for the initial model and the updated model, the product of the updated stiffness matrix \(\mathbf{K}_{mt}\) and the measured displacement vector \(\mathbf{x}_{mt}\) should equal to the concentrated force vector \(\mathbf{F}_{at}\). Note that this principle is different from that presented in literature [5] by [PERSON] et al., where a displacement equation error function is defined and minimized to determine the updating parameters. Using Equation (3), the following equation can be obtained:  
\[\mathbf{K}_{at}\mathbf{x}_{at}=(\left.\mathbf{K}_{mt}\right|_{\alpha=0}+\sum_ {i=1}^{n}\alpha_{i}\frac{\partial\mathbf{K}_{mt}}{\partial\alpha_{i}}\right|_ {\alpha_{i}=0})\mathbf{x}_{mt} \tag{9}\]  
where \(\left.\mathbf{K}_{mt}\right|_{\alpha=0}=\mathbf{K}_{at}\) the translational part of the stiffness matrix of the initial simulation model.