## 3 FEM Computations of the Bridge Natural Values  
The finite element method (FEM) is a method derived from the variational principle. The actual course of the displacement components is replaced by elementary substitution functions in the form of polynomials. The computational model created in the spirit of FEM is a model with a finite number of degrees of freedom. The equations have the meaning of the generalized conditions of an equilibrium at the nodes. In a dynamic task, inertial forces also enter into conditions of equilibrium [9]. If the vector of unknown displacement components in the nodes is denoted as \(\{r\}\), then in the case of an undamped system, the equations have the form:  
Figure 1: Cross-section of the bridge. Dimensions in mm. \[[M]\cdot\left\{r\right\}+[K]\cdot\left\{r\right\}=\left\{0\right\} \tag{1}\]  
where \([M]\) is the global mass matrix of the structure (in our case it will be diagonal) and \([K]\) is the global stiffness matrix of the structure. The solution of Equation (1) is sought in the form:  
\[\left\{r\right\}=\left\{r_{(j)}\right\}\cdot\sin(\omega_{(j)}\cdot t) \tag{2}\]  
Substituting (2) into (1) gives a set of frequency equations in the form:  
\[\left([K]-\omega_{(j)}^{2}\cdot[M]\right)\cdot\left\{r_{(j)}\right\}=\left\{0\right\} \tag{3}\]  
The condition for the solvability of the system of Equation (3) is:  
\[\det([K]-\omega_{(j)}^{2}\cdot[M])=0 \tag{4}\]  
If the matrices \([K]\) and \([M]\) are of order \(n\cdot n\), it is possible to calculate \(n\) natural angular frequencies \(\omega_{(j)}\) and \(n\) natural modes \(\left\{r_{(j)}\right\}\) from Equation (3). If the matrix \([M]\) is diagonal, it is possible to transform Equation (3) into the eigenvalue problem of matrices in the form:  
\[\left([K_{r}]-\lambda_{(j)}\right)\cdot\left\{r_{(j)}\right\}=\left\{0\right\} \tag{5}\]  
where:  
\[\lambda_{(j)}=\omega_{(j)}^{2} \tag{6}\]  
and:  
\[[K_{r}]=[M]^{-1}\cdot[K]. \tag{7}\]  
The eigenvalues \(\lambda_{(j)}\) of the matrix \([K_{r}]\) represent the square of the natural frequencies \(\omega_{(j)}^{2}\), and the eigenvectors \(\left\{r_{(j)}\right\}\) of the matrix \([K_{r}]\) represent the natural modes. The computational model of the structure was created from slab-wall elements. A quadrilateral, eight-node, Mindlin element was used to create the computational model [10]. The element has nodes in the corners of the rectangle and the centers of the sides. The center node has displacement vector component \(\left\{u\right\}=[u\),\(v\),\(w]^{T}\) as the unknown, and the corner nodes have displacement vector component \(\left\{u\right\}=[u\),\(v\),\(w]^{T}\) and rotation vector component \(\left\{\varphi\right\}=[\varphi_{(x)}\), \(\varphi_{(y)}\), \(\varphi_{(z)}]^{T}\) as unknowns. The whole structure was assembled from six types of elements. The total number of elements is 812. The cross-sectional dimensions of the model are shown in Figure 2. The number of degrees of freedom of the model is 8259. The dimensions of the individual types of elements and their mass characteristics are in Table 1. Material characteristics: modulus of elasticity \(E\) = 3.85 \(\times\) 10\({}^{10}\) N/m\({}^{2}\), [PERSON]'s ratio \(v\) = 0.15. The first 10 natural frequencies are arranged in Table 2, and the natural modes are shown in \[[M]\cdot\left\{r\right\}+[K]\cdot\left\{r\right\}=\left\{0\right\} \tag{1}\]0 The serial number of the natural mode is denoted by \(j\).