A water delivery system can be gradually away from the normal state of the canal operation in northern China in winter. Water diversion projects can be launched to meet the harsh requirement on the large and medium-sized city's residents living water and industrial water consumption, as the guaranteed rates increased. However, the existing mechanical model of the canal with the water delivery cannot fully consider the difference between the freezing and underwater non-freezing areas in winter, particularly for the continuity of soil. It is a high demand to combine the difference in shear stiffness between frozen soil and non-frozen soil. Pasternak model with two parameters can be expected to consider the soil continuity in the freezing area, whereas, the traditional Winkler model can be adopted in the non-freezing area. By contrast, the Winkler model with only a few parameters can be easily calculated for definite physical significance. In this study, the combined Winkler-Pasternak model was proposed for the frost-heaving mechanical analysis of the trapezoidal canal with water delivery in winter. Taking a trapezoidal canal with water delivery in winter in the Manas River basin in Xinjiang of western China as a prototype, the real normal frost-heaving amount, and subsidence deformation of the canal lining plate were also calculated using the improved model, Winkler and Pasternak model. Then, the bending moment was calculated with the upper surface stress of each section of the lining plate. The results indicate that the lining plate was divided into three parts of frost heave, subsidence and frost heave-subsidence transition section. Three models better represented the basic change trend of normal displacement of the lining plate. Meanwhile, the improved model was in better agreement with the measurement. A comparative analysis showed that the improved model was much more accurate than the traditional Winkler model in the freezing area, whereas, some errors but trivial differences were compared with the Pasternak model in the non-freezing area. Furthermore, the improved model shared the excellent performance of the Winkler model, such as simple calculation, clear physical meaning, and few required parameters. The bending moment of sections decreased rapidly in the freezing area with the increase of groundwater depth. The risk of frost-heaving damage to the canal lining plate was dramatically reduced at the groundwater level by appropriate drainage measures. The position in the frozen soil area of the lining plate easy to crack was located at 10.0%~23.3% of the lining plate length away from the waterline. The optimal operation conditions were adopted in the ice-free water delivery of the trapezoidal canal in winter. The improved model can be also applied to the ever-increasingly common situation of water delivery with the ice cover. The finding can provide a strong reference to design the frost heave resistance of the trapezoidal canal with water delivery in winter.

Frost-heaving resistance of canals can be designated to evaluate the control parameters in cold regions. As proposed in the national standard of SL23-2006 "Design Code for Frost Heave Resistance of Canal System Engineering", the frost-heaving deformation of the canal concrete lining structure can be expected to readily monitor and control in practice. However, the specification cannot provide the specific calculation of uneven frost-heaving deformation in the canal concrete lining. This study aims to facilitate the accurate and practical calculation for the frost-heaving deformation of concrete lining structures under uneven frost-heaving deformation conditions. The fluctuation of groundwater levels was considered across different points along the canal cross-section, together with the continuous deformation within the canal foundation soil. A calculation model was developed for the frost-heaving deformation and internal force of trapezoid concrete lining canal in cold regions under differential frost heave conditions, according to the two-parameter foundation beam of frozen soil. Taking the main trunk canal of Jinghui in Gansu Province as an example, the presented model degenerated into the Winkler model under uniform frost heave conditions. Comparative analysis showed that the traditional Winkler model was regarded as a special case within the proposed model. Some measuring points were simultaneously considered for the shady slope, sunny slope, and bottom of the canal. The Root Mean Square Error (RMSE) between the presented model and observed values was determined to be 0.23, with the Mean Relative Error (MRE) of 4.82%. Taking a trapezoidal concrete lining canal in the Tarim Irrigation area of Xinjiang Uygur Autonomous Region as the prototype, the frost-heaving deformation of the concrete lining structure was calculated at the same time. The results show that the better mechanical performance of the concrete lining canal was achieved in the improved model considering the frozen soil-structure interaction and diffusion of shear force in the frozen soil foundation under uneven frost heave conditions, compared with the material mechanics and traditional Winkler model. The calculated values of the improved model were more consistent with the observation, indicating the rationality and applicability of the model. The engineering example was combined to solve the improved model. The first was the conventional analytical, while the second involved the power series. Among them, the various distributions of free frost heave deformation were expanded into the power series using the comparison coefficient. A special solution with a homogeneous solution was then obtained to introduce the boundary conditions. The general solution was derived for the original control differential equation. The MRE between the frost-heaving deformation was calculated by the power series. Among them, the observed value was 4.31% with an RMSE of 0.17, indicating a reasonable calculation. The power series was utilized to represent the natural frost-heaving displacement of canal foundation soil. Therefore, the improved model effectively enhanced the applicability for the various differential frost-heaving modes in foundation-frozen soil. The mechanical response of frost heave was found in the trapezoidal concrete lining canal under differential or uniform frost-heaving deformation. Then, some implication was gained to clarify the influence of transverse groundwater replenishment on the mechanical behavior of concrete lining structures under the frost-heaving deformation of foundation soil. In addition, the improved model can be expected to effectively capture the non-uniform and asymmetric distribution of tangential freezing force at the contact interface, due to the non-equilibrium axial force on both ends of the bottom plate that is caused by the difference between shady and sunny slopes. The finding can provide an effective computational model for the frost heave mechanics of trapezoidal canals in cold regions, especially on differential frost heave conditions.