# A Statistical Model Updating Method of Beam Structures with Random Parameters under Static Load
## 2 New Statistical Model Updating Method
### Solution of Statistical Model Updating Equation  
Assuming that the measurement errors of the beam structure under static load are random, the measured displacement vector of the actual structure can be expressed as:  
\[\mathbf{x}_{mt}=\mathbf{x}_{mt0}+\sum_{i=1}^{l}\beta_{i}\mathbf{x}_{mti} \tag{11}\]  
where \(l\) is the number of random factors affecting the measurement, and equals to 1 given that the measurement errors are completely dependent. \(\mathbf{x}_{mt0}\) is the mean of the measured displacement vector, \(\beta_{i}\) is the independent random variable, and \(\mathbf{x}_{mti}\) is the adjoint vector of the random variable \(\beta_{i}\). Using the multivariable power series to expand the element updated factors \(\alpha_{i}\) (\(i=1,\,\ldots,\,n\)), one can have  
\[\alpha_{i}=\alpha_{i0}+\sum_{j=1}^{l}\alpha_{ij}\beta_{j}+\sum_{j=1}^{l}\sum_{ k=1}^{j}\alpha_{ijk}\beta_{j}\beta_{k}+\sum_{j=1}^{l}\sum_{k=1}^{j}\sum_{p=1}^{k} \alpha_{ijkp}\beta_{j}\beta_{k}\beta_{p}+... \tag{12}\]  
where \(\alpha_{i0}\), \(\alpha_{ij}\), \(\alpha_{ijk}\) and \(\alpha_{ijkp}\) are respectively the zero-order to third-order of unknown expansion coefficients of the multivariable power series (12). Substituting Equation (12) for Equation (10) leads to:  
\[\mathbf{K}_{at}(\mathbf{x}_{at}-(\mathbf{x}_{mt0}+\sum_{i=1}^{l}\beta_{i} \mathbf{x}_{mti}))=\sum_{i=1}^{n}\left(\alpha_{i0}+\sum_{j=1}^{l}\alpha_{ij} \beta_{j}+\sum_{j=1}^{l}\sum_{k=1}^{j}\alpha_{ijk}\beta_{j}\beta_{k}+\cdots \right)\mathbf{K}_{ti}(\mathbf{x}_{mt0}+\sum_{i=1}^{l}\beta_{i}\mathbf{x}_{ mti}) \tag{13}\]Obviously, Equation (13) is a random algebraic equation with unknown expansion coefficients \(\alpha_{i0}\), \(\alpha_{ij}\), \(\alpha_{ijk}\), etc. A high-order perturbation technique, which has been successfully used to determine the random frequencies of a large span cable-stayed bridge in the literature [27], is suggested to solve Equation (13). Using the high-order perturbation technique, the expansion coefficients \(\alpha_{i0}\), \(\alpha_{ij}\), \(\alpha_{ijk}\) and \(\alpha_{ijkp}\), which correspond to the power polynomials 1, \(\beta_{j}\), \(\beta_{j}\beta_{k}\) and \(\beta_{j}\beta_{k}\beta_{p}\) respectively, can be recursively determined. At first, for the zero-order polynomial term, there is:  
\[\alpha_{i0}=\mathbf{A}_{0}^{-1}(\mathbf{K}_{at}\mathbf{x}_{at}-\mathbf{K}_{at }\mathbf{x}_{mt0}) \tag{14}\]  
where \(\mathbf{A}_{0}=[\mathbf{K}_{t1}\mathbf{x}_{mt0},\mathbf{K}_{t2}\mathbf{x}_{ mt0},...,\mathbf{K}_{tn}\mathbf{x}_{mt0}]\). Next, for the first-order power polynomial \(\beta_{j}\), one has:  
\[\alpha_{ij}=-\mathbf{A}_{0}^{-1}(\mathbf{K}_{at}\mathbf{x}_{mti}+\mathbf{A}_{ 1}\alpha_{i0}) \tag{15}\]  
where \(\mathbf{A}_{1}=[\mathbf{K}_{t1}\mathbf{x}_{mtj},\mathbf{K}_{t2}\mathbf{x}_{ mtj},...,\mathbf{K}_{tn}\mathbf{x}_{mtlj}]\). Then, considering the second-order power polynomial, \(\beta_{j}\beta_{k}\) results in:  
\[\alpha_{ijk}=-\mathbf{A}_{0}^{-1}\mathbf{A}_{2}\alpha_{ij} \tag{16}\]  
where \(\mathbf{A}_{2}=[\mathbf{K}_{t1}\mathbf{x}_{mtk},\mathbf{K}_{t2}\mathbf{x}_{ mtk},...,\mathbf{K}_{tn}\mathbf{x}_{mtk}]\). For the third-order power polynomial \(\beta_{j}\beta_{k}\beta_{p}\), the expansion coefficient \(\alpha_{ijkp}\) is obtained as:  
\[\alpha_{ijkp}=-\mathbf{A}_{0}^{-1}\mathbf{A}_{3}\alpha_{ijk} \tag{17}\]  
where \(\mathbf{A}_{3}=[\mathbf{K}_{t1}\mathbf{x}_{mtp},\mathbf{K}_{t2}\mathbf{x}_{ mtp},...,\mathbf{K}_{tn}\mathbf{x}_{mtp}]\). In the same way, the fourth- to the higher-order of expansion coefficients can be determined. Taking into account the balance between the accuracy and the efficiency, up to the fourth-order of coefficients are used in this paper. Further, by the power polynomial expansion solved, the statistics of the element updated factors can be obtained easily. Note that many random methods, such as the Monte Carlo simulation methods, the spectral stochastic finite element methods (SSFEM), etc., can be used to solve the statistical model updating Equation (10).