Application of reaction force coefficient in the mechanical model for the elastic foundation beam of canal lining during frost heave

Canal foundation soil is generally characterized by nonlinear deformation during the frost heaving process. In this study, a Duncan-Chang constitutive model was established considering the confining pressure and temperature, according to the frozen soil triaxial test. A numerical simulation was conducted to establish the calculation formula for the frost heave reaction force coefficient with the constrained frost heave change, particularly referring to the bed coefficient in the indoor triaxial test. The balance differential equation of the elastic foundation beam was discretized to adapt to the change of frost heave reaction coefficient using the finite difference method. Since the frost heave reaction coefficient greatly varied at different points of lining, the model increased significantly the much more constants than before. Analytical solutions were also applied to verify the new model. In addition, the difference was also determined, when the frost heave reaction force coefficients were variable and constant, particularly for the frost heave mechanical response of trapezoidal canal lining. The results showed that the frost heave reaction force increased by 0.15 MPa on average, while the frost heave reaction force coefficient decreased by 1.12 MPa/m on average at the same freezing temperature for every 1 cm increase in the constrained frost heave. The frost heave reaction force increased by 0.21 times on average for every 5°C decrease at freezing temperature under the same restricted frost heave amount. As such, the hyperbolic function was used to represent this change suitable for engineering applications. In slope and canal bottom lining slabs, the maximum frost heave reaction force calculated by the constant frost heave reaction force coefficient was 1.43 times the variable, and the maximum bending moment was 1.12 times the variable on average. Therefore, the nonlinear deformation of frozen soil was not considered to avoid a larger value than before, if the frost heave reaction coefficient was constant. Moreover, the significant influence of the frost heave reaction coefficient was closely related to the length of the lining slab. The maximum bending moment gradually increased and then became stable, while the position of the maximum shifted to the tip of the slope, as the length of the slope lining slab increased. By contrast, the position of the maximum bending moment changed from the middle to two sides, and the midpoint bending moment first increased and then decreased, as the length of the lining slab at the bottom of the canal increased. This finding can provide strong technical support to the design of large-scale trapezoidal canal lining. The frost heave reaction force coefficient was relatively smaller than the actual value, due mainly to the model without considering water migration well. At the same time, it is also necessary to consider the effect of the separation of lining board from the foundation, and the freezing shrinkage of lining board, on the bending in the future research.