About the approximation of the tunnel lining deformations according to the measurement results

The article deals with the approximation of the displacements of the points of the tunnel lining contour, obtained, for example, by geodetic survey. In the future, the processed data can be used to calculate the internal forces in the tunnel lining, which will help in monitoring the technical condition of the bearing structures of underground structures. In the article, calculation formulas have been obtained that allow using the measured displacements of the reference points to find the displacements of any point of the lining. Displacements are determined by approximating deformations with periodic cubic splines. To determine the coefficients of splines, it is proposed to apply the common least squares method. A method for calculating the weighting coefficients of basic functions for a spline with an arbitrary step of the nodes is presented. As an intermediate result, analytical dependences were obtained between the Cartesian coordinates of the tunnel lining points and the arc length. In this case, the design data on the radii of curvature of the lining sections and the coordinates of the points of section conjugation were used. It is assumed that the data obtained will increase the information content of the control and can be used to monitor its technical condition of underground structures at all stages of its life cycle.