The warm and ice-rich frozen soil is characterized by high unfrozen water content, low shear strength and large compressibility, which is unreliable to meet the stability requirements of engineering infrastructures an...The warm and ice-rich frozen soil is characterized by high unfrozen water content, low shear strength and large compressibility, which is unreliable to meet the stability requirements of engineering infrastructures and foundations in permafrost regions. In this study, a novel approach for stabilizing the warm and ice-rich frozen soil with sulphoaluminate cement was proposed based on chemical stabilization. The mechanical behaviors of the stabilized soil, such as strength and stress-strain relationship, were investigated through a series of triaxial compression tests conducted at -1.0℃, and the mechanism of strength variations of the stabilized soil was also explained based on scanning electron microscope test. The investigations indicated that the strength of stabilized soil to resist failure has been improved, and the linear Mohr-Coulomb criteria can accurately reflect the shear strength of stabilized soil under various applied confining pressure. The increase in both curing age and cement mixing ratio were favorable to the growth of cohesion and internal friction angle. More importantly, the strength improvement mechanism of the stabilized soil is attributed to the formation of structural skeleton and the generation of cementitious hydration products within itself. Therefore, the investigations conducted in this study provide valuable references for chemical stabilization of warm and ice-rich frozen ground, thereby providing a basis for in-situ ground improvement for reinforcing warm and ice-rich permafrost foundations by soil-cement column installation.展开更多
Affected by climate warming and anthropogenic disturbances, the thermo-mechanical stability of warm and ice-rich frozen ground along the Qinghai-Tibet engineering corridor(QTEC) is continuously decreased, which may de...Affected by climate warming and anthropogenic disturbances, the thermo-mechanical stability of warm and ice-rich frozen ground along the Qinghai-Tibet engineering corridor(QTEC) is continuously decreased, which may delay the construction of major projects in the future. In this study, based on chemical stabilization of warm and icerich frozen ground, the soil-cement column(SCC) for ground improvement was recommended to reinforce the foundations in warm and ice-rich permafrost regions. To explore the validity of countermeasures mentioned above, both the original foundation and the composite foundation consisting of SCC with soil temperature of -1.0℃ were prepared in the laboratory, and then the plate loading tests were carried out. The laboratory investigations indicated that the bearing capacity of composite foundation consisting of SCC was higher than that of original foundation, and the total deformation of original foundation was greater than that of composite foundation, meaning that overall stability of foundation with warm and ice-rich frozen soil can be improved by SCC installation. Meanwhile, a numerical model considering the interface interaction between frozen soil and SCC was established for interpretating the bearing mechanism of composite foundation. The numerical investigations revealed that the SCC within composite foundation was responsible for the more applied load, and the applied load can be delivered to deeper zone in depth due to the SCC installation, which was favorable for improving the bearing characteristic of composite foundation. The investigations provide the valuable guideline for the choice of engineering supporting techniques to major projects within the QTEC.展开更多
In permafrost regions with warm frozen soil,subgrade thaw-collapse phenomenon commonly occurs,facing thaw collapse problems of the existed frozen soil subgrade,thus it is difficult to use traditional methods such as a...In permafrost regions with warm frozen soil,subgrade thaw-collapse phenomenon commonly occurs,facing thaw collapse problems of the existed frozen soil subgrade,thus it is difficult to use traditional methods such as active cooling and passive protection technology to stabilize the existed warm frozen soil subgrade.This study derives a novel stabilizer method,a long-short(L-S)cement-mixed batter pile composite foundation to stabilize the existed warm frozen soil subgrade.To solve the thawcollapse problems in warm frozen soil subgrade,high water content and large compressibility characteristics were compared between soft soil and warm frozen soils.Theoretical analysis of heat conduction and numerical simulation of finite element model were used to study the freeze–thaw process and evaluate the stabilized effects of the L-S cement-mixed batter piles on the warm frozen soil foundation of the Qinghai-Tibet Highway.Furthermore,the thaw process and mechanical properties of foundation and piles were analyzed by introducing the hydration heat factor in the thermodynamic control equation.The results indicate that the thawing displacement of the existed warm frozen soil subgrade was reduced owing to the“support”and“grasp”effects of the L-S cement-mixed batter piles on the surrounding soil.The composite ground formed by strengthening the warm frozen ground with batter piles could considerably improve the bearing capacity of the existed warm frozen ground,effectively restrain the deformation of the upper embankment,and improve the strength of the ground.The analysis can provide method for the construction design of cement mixing batter pile foundation in cold regions.展开更多
A rate-dependent constitutive model for saturated frozen soil is vital in frozen soil mechanics,especially when simultaneously describing the nonlinearity,dilatancy and strain-softening characteristics.The distributio...A rate-dependent constitutive model for saturated frozen soil is vital in frozen soil mechanics,especially when simultaneously describing the nonlinearity,dilatancy and strain-softening characteristics.The distribution of the non-uniform strain rate of saturated frozen soil at the meso-scale due to the local icecementation breakage is described by a newly binary-medium-based homogenization equation.Based on the field-equation-based approach of the meso-mechanics theory,the interaction expression of the strain rate at macro-and meso-scale is derived,which can give the strain rate concentration tensor at different crushed degrees.With the thermodynamics and empirical assumption,a breakage ratio in the rate-dependent form is determined.This overcomes the limitations of the existing binary-medium-based models that are difficult to simulate rate-dependent mechanical response.Based on these assumptions,a newly binary-medium-based rate-dependent model is proposed considering both the ice bond breakage and material composition characteristics of saturated frozen soil.The proposed constitutive model has been validated by the test results on frozen soils with different temperatures and strain rates.展开更多
The acquisition of spatial-temporal information of frozen soil is fundamental for the study of frozen soil dynamics and its feedback to climate change in cold regions.With advancement of remote sensing and better unde...The acquisition of spatial-temporal information of frozen soil is fundamental for the study of frozen soil dynamics and its feedback to climate change in cold regions.With advancement of remote sensing and better understanding of frozen soil dynamics,discrimination of freeze and thaw status of surface soil based on passive microwave remote sensing and numerical simulation of frozen soil processes under water and heat transfer principles provides valuable means for regional and global frozen soil dynamic monitoring and systematic spatial-temporal responses to global change.However,as an important data source of frozen soil processes,remotely sensed information has not yet been fully utilized in the numerical simulation of frozen soil processes.Although great progress has been made in remote sensing and frozen soil physics,yet few frozen soil research has been done on the application of remotely sensed information in association with the numerical model for frozen soil process studies.In the present study,a distributed numerical model for frozen soil dynamic studies based on coupled water-heat transferring theory in association with remotely sensed frozen soil datasets was developed.In order to reduce the uncertainty of the simulation,the remotely sensed frozen soil information was used to monitor and modify relevant parameters in the process of model simulation.The remotely sensed information and numerically simulated spatial-temporal frozen soil processes were validated by in-situ field observations in cold regions near the town of Naqu on the East-Central Tibetan Plateau.The results suggest that the overall accuracy of the algorithm for discriminating freeze and thaw status of surface soil based on passive microwave remote sensing was more than 95%.These results provided an accurate initial freeze and thaw status of surface soil for coupling and calibrating the numerical model of this study.The numerically simulated frozen soil processes demonstrated good performance of the distributed numerical model based on the coupled water-heat transferring theory.The relatively larger uncertainties of the numerical model were found in alternating periods between freezing and thawing of surface soil.The average accuracy increased by about 5%after integrating remotely sensed information on the surface soil.The simulation accuracy was significantly improved,especially in transition periods between freezing and thawing of the surface soil.展开更多
The correct determination of thermal parameters,such as thermal conductivity and specific heat of soil during freezing,is the most important and basic problem for the construction of an appropriate freezing method.In ...The correct determination of thermal parameters,such as thermal conductivity and specific heat of soil during freezing,is the most important and basic problem for the construction of an appropriate freezing method.In this study,a calculation model of three stages of soil temperature was established.At the unfrozen and frozen stages,the specific temperatures of dry soil,water,and ice are known.According to the principle of superposition,a calculation model of unfrozen and frozen soils can be established.Informed by a laboratory experiment,the latent heat of the adjacent zone was calculated for the freezing stage based on different water contents in the temperature section.Both the latent and specific heat of water,ice,and particles were calculated via superposition of the weight percentage content.A calculation model of the specific heat of the freezing stage was built,which provides both guidance and theoretical basis for the calculation of the specific heat of frozen soil.展开更多
Numerical simulation is known as an effective method for mechanical properties during frozen soil excavation.In order to reveal the development of cutting force,effective stress and cutting fragments in frozen silt du...Numerical simulation is known as an effective method for mechanical properties during frozen soil excavation.In order to reveal the development of cutting force,effective stress and cutting fragments in frozen silt during the cutting process,we introduce an explicit finite element program LS-DYNA to establish a two-dimensional numerical model of the frozen soil cut.We also use the Holmquist-Johnson-Cook(HJC)damage constitutive model for simulating the variation of soil mechanical properties according to the strong dependence between the cutting tool and frozen silt during the process with different cutting depths,angles and velocities.Meanwhile,a series of experimental results are acquired of frozen silt cutting to prove the application of the HJC model during simulation of cutting force variations.The result shows that the cutting force and fragment size are strongly influenced by cutting depths and cutting velocities increased,and the maximum effective stress at points where the tool contacts frozen soil during the cutting process.In addition,when the cutting angle is 52°,the cutting force is the smallest,and the cutting angle is optimum.Thus,the prediction of frozen soil mechanical properties on the cutting process by this model is conducive to selecting machinery equipment in the field.展开更多
In view of the cumbersome and often untimely process of manual collection and observation of frozen soil data parameters,and the damage caused to dams by frost heaving of frozen soil,a remote monitoring and an early w...In view of the cumbersome and often untimely process of manual collection and observation of frozen soil data parameters,and the damage caused to dams by frost heaving of frozen soil,a remote monitoring and an early warning model for frozen soil in dam areas was presented.The Pt100 temperature sensors and JM seam gauges were used as measurement tools in the system.The sensor layout was designed,based on the actual situation in the monitoring area.A 4G network was used for wireless transmission to monitor frozen soil data in real time.BP neural network was used to predict the parameters of frozen soil.After analysis,four factors including the average temperature of frozen soil,the type of frozen soil,the artificial upper limit of frozen soil and the building construction time were selected to establish an early warning model using fuzzy reasoning.The experimental results showed that the early warning model could reflect the influence on dam buildings of frost heaving and sinking of frozen soil,and provided technical support for predicting the hazard level.展开更多
A series of directional shear tests on remolded frozen soil was carried out at 10°C by using a hollow cylinder apparatus to study failure criterion under a directional shear-stress path.Directional shear tests we...A series of directional shear tests on remolded frozen soil was carried out at 10°C by using a hollow cylinder apparatus to study failure criterion under a directional shear-stress path.Directional shear tests were conducted at five shear rates(10,20,30,40,and 50 kPa/min)and five intermediate principal stress coefficients(b=0,0.25,0.5,0.75,and 1),with the mean principal stress(p=4.5 MPa)kept constant.The results show that the torsional strength and the generalized strength both increase with the increase of the shear rates.According to the failure modes of frozen soil under different shear rates,the specimens present obvious plastic failure and shear band;and the torsional shear component dominates the failure modes of hollow cylindrical specimens.A shear rate of 30 kPa/min is chosen as the loading rate in the directional shear tests of frozen soil.The shape of the failure curve in theπplane is dependent on the directional anglesαof the major prin cipal stress.It is reasonable to use the strain-hardening curves to define the deviatoric stress value atγg=15%(generalized shear strain)as the failure criterion of frozen soil under a directional shear-stress path.展开更多
Soil slope stability in seasonally frozen regions is a challenging problem for geotechnical engineers.The freezethaw process of soil slope caused by the temperature fluctuation increases the difficulty in predicting t...Soil slope stability in seasonally frozen regions is a challenging problem for geotechnical engineers.The freezethaw process of soil slope caused by the temperature fluctuation increases the difficulty in predicting the slope stability because the soil property is influenced by the freeze-thaw cycle.In addition,the frozen soil,which has ice crystal,ice lens and experienced freeze-thaw process,could present stronger heterogeneity.Previous research has not investigated the combined effect of soil heterogeneity and freeze-thaw cycle.This paper studies the influence of soil heterogeneity on the stability of frozen soil slope under freeze-thaw cycles.The local average subdivision(LAS)is utilized to model the soil heterogeneity.A typical slope geometry has been chosen and analysed as an illustrative example and the strength reduction method is used to calculate the factor of safety(FOS)of slope.It has been found that when the temperature is steady,the FOS of the frozen soil slope is influenced by the spatial variability of the thermal conductivity,but the influence is not significant.When the standard deviation and the SOF of the thermal conductivity increase,the mean of the FOS is equal to the FOS of the homogeneous case and the standard deviation of the FOS also increases.After the frozen soil goes through freeze-thaw process,the FOS of the frozen soil slope decreases due to the reduction in the cohesion and the internal friction angle caused by the freeze-thaw cycles.Furthermore,the decreasing ratio of the FOS becomes more scattered after the 5th freeze-thaw cycle compared to that of the FOS after the 1st freeze-thaw cycle.The larger variability of the FOS could induce inaccuracy in the prediction of the frozen soil slope stability.展开更多
The vibration of underground or buried piping during construction and long-term operation causes secondary disasters,and becomes more complex when tubes are buried in cold regions.The interface between saturated froze...The vibration of underground or buried piping during construction and long-term operation causes secondary disasters,and becomes more complex when tubes are buried in cold regions.The interface between saturated frozen soil and lining is regarded as a thin spring-like layer whose thickness could be negligible.In this paper,the dynamic response of saturated frozen soil is studied in frequency domain by using the Helmholtz composition and Fourier transform to obtain analytical solutions of the radial and axial displacement,as well as expressions of the stiffness coefficient(Kr)and damping coefficient(Cr)of the spring-like interface.Numerical results indicate that Krand Crare related to physical properties of the lining and its surrounding soil,and the coefficients of the springlike model could be changed by adjusting lining parameters to improve structure stability under the same load conditions.Also,the viscoelastic contact surface of the spring-like model is considered to have less effect on the surrounding soil than that when the lining has complete contact with the soil under load.The degree of soil freezing significantly affects the axial and radial displacement of the soil when the interface between lining and unsaturated frozen soil is taken into consideration.展开更多
The freezing of soil containing a liquid is a complex transient heat conduction problem involving phase change and release or absorption of latent heat.Existing efforts have essentially focused on theoretical research...The freezing of soil containing a liquid is a complex transient heat conduction problem involving phase change and release or absorption of latent heat.Existing efforts have essentially focused on theoretical research and numerical simulations.In the present study,the problem is approached from an experimental point of view using the so-called“freezing model test”method.In particular,in order to establish a precise relationship between the model and the prototype,a temperature similarity criterion is derived using the condition of an equal number of Kosovitch.Similarity is also established with respect to other aspects.A similarity criterion for the water field is determined on the basis of relevant partial differential equations.Analogous criteria for the stress field and load are derived using an elastic model.The validity of this approach is experimentally verified.The research results provide a practical and reasonable method for calculating the parameters for preparing model soils.They also constitute a theoretical basis and a technical support for the design and implementation of a water-heat-force similarity coupled framework.展开更多
Seasonally frozen soil in alpine and subalpine zones in the mountains of Qinghai-Tibetan Plateau is particularly sensitive to global climate change. Therefore, a better understanding of the thermal properties of froze...Seasonally frozen soil in alpine and subalpine zones in the mountains of Qinghai-Tibetan Plateau is particularly sensitive to global climate change. Therefore, a better understanding of the thermal properties of frozen soil is crucial for predicting the responses of frozen soils to soil warming. In this study, thermal properties of frozen soil with different moisture contents under subzero temperature (0°C - 20°C) in an alpine forest in western Sichuan were analyzed by KD<sub>2</sub> Pro in its cooling and heating processes, respectively. Our results reveal that the soil apparent volumetric specific heat capacity (C<sub>v</sub>) and apparent thermal conductivity (K) under the same water content show similar response patterns to changing temperature lower than -2°C in both heating and cooling processes. Moreover, ice content of frozen soils can be well predicted by Logistic model in cooling and heating processes. The C<sub>v</sub> and K tend to increase along with increasing soil moisture contents. Remarkably, asymptotic characters of the value of C<sub>v</sub> and K are at the vicinity of the initial temperature of phase transitions, indicating that both C<sub>v</sub> and K are particularly sensitive to changing soil temperature at the range of -2°C to 0°C. Therefore, the widely distributed frozen soil layers with temperature above -2°C in alpine and subalpine zones over Qinghai-Tibetan Plateau are susceptible to the observed climate warming during cold season.展开更多
The split Hopkinson pressure bar (SHPB) method is used to investigate the dynamic behavior of the artificial frozen soil under the nearly uniaxial strain and uniaxial stress conditions. The tests are conducted at the ...The split Hopkinson pressure bar (SHPB) method is used to investigate the dynamic behavior of the artificial frozen soil under the nearly uniaxial strain and uniaxial stress conditions. The tests are conducted at the temperatures of 3℃, 8℃, 13℃, 17℃, 23℃, and 28℃ and with the strain rates from 900 s-1 to 1 500 s-1 . The nearly uniaxial stress-strain curves exhibit an elastic-plastic behavior, whereas the uniaxial stress-strain curves show a brittle behavior. The compressive strength of the frozen soil exhibits the positive strain rate and negative temperature sensitivity, and the final strain of the frozen soil shows the positive strain rate sensitivity. The strength of the frozen soil under the nearly uniaxial strain is greater than that under the uniaxial stress. After the negative confinement tests, the specimens are compressed, and the visible cracks are not observed. However, the specimens are catastrophically damaged after the uniaxial SHPB tests. A phenomenological model with the thermal sensitivity is established to describe the dynamic behavior of the confined frozen soil.展开更多
Warm and ice-rich frozen soil(WIRFS) exhibits lower shear strength due to the weak binding forces between soil particles and ice crystals. To enhance the strength of WIRFS, frozen soil was treated separately with Port...Warm and ice-rich frozen soil(WIRFS) exhibits lower shear strength due to the weak binding forces between soil particles and ice crystals. To enhance the strength of WIRFS, frozen soil was treated separately with Portland, Phosphate, Sulphoaluminate, Portland-Phosphate and PortlandSulphoaluminate cements. After the samples were cured under -1.0°C for 7 days, the microscopic pore distribution characteristics and the macro-mechanical properties of WIRFS were investigated using mercury intrusion porosimetry(MIP), scanning electron microscopy(SEM) and unconfined compressive strength(UCS) tests. To quantitatively analyze the laws of pore-size transformation and the variation of Hausdorff volumetric fractal dimensions for pre-and post-treated WIRFS, the CURVEEXTRACT and Image-Pro Plus(IPP) image analysis system has been developed for analysing SEM images of the soil samples. Statistics of the pore-area dimension and pore-volume dimension were calculated. The results reveal that the cement-based treatment of WIRFS can improve the cementation fill of soil pores and the bond forces between soil particles. There is an evident correlation between the microstructure characteristics and the mechanical properties of the treated WIRFS. As the fractal dimensions of pore-area decrease, the unconfined compressive strength of cement-treated WIRFS increases significantly. In contrast, as the fractal dimensions of pore-volume increases, the unconfined compressive strength decreases remarkably.展开更多
Using newly developed dynamic shearing devices, the dynamic shearing strength of frozen soil-concrete interface was studied experimentally. By placing concrete blocks in the lower half of the shear box and frozen soil...Using newly developed dynamic shearing devices, the dynamic shearing strength of frozen soil-concrete interface was studied experimentally. By placing concrete blocks in the lower half of the shear box and frozen soil sample in the upper part, a series of dynamic shear tests on their interfaces were carried out. The obtained results are summarized and the main influencing factors are revealed.展开更多
In almost all frozen soil models used currently, three variables of temperature, ice content and moisture content are used as prognostic variables and the rate term, accounting for the contribution of the phase change...In almost all frozen soil models used currently, three variables of temperature, ice content and moisture content are used as prognostic variables and the rate term, accounting for the contribution of the phase change between water and ice, is shown explicitly in both the energy and mass balance equations. The models must be solved by a numerical method with an iterative process, and the rate term of the phase change needs to be pre-estimated at the beginning in each iteration step. Since the rate term of the phase change in the energy equation is closely related to the release or absorption of the great amount of fusion heat, a small error in the rate term estimation will introduce greater error in the energy balance, which will amplify the error in the temperature calculation and in turn, cause problems for the numerical solution convergence. In this work, in order to ?rst reduce the trouble, the methodology of the variable transformation is applied to a simpli?ed frozen soil model used currently, which leads to new frozen soil scheme used in this work. In the new scheme, the enthalpy and the total water equivalent are used as predictive variables in the governing equations to replace temperature, volumetric soil moisture and ice content used in many current models. By doing so, the rate terms of the phase change are not shown explicitly in both the mass and energy equations and its pre-estimation is avoided. Secondly, in order to solve this new scheme more functionally, the development of the numerical scheme to the new scheme is described and a numerical algorithm appropriate to the numerical scheme is developed. In order to evaluate the new scheme of the frozen soil model and its relevant algorithm, a series of model evaluations are conducted by comparing numerical results from the new model scheme with three observational data sets. The comparisons show that the results from the model are in good agreement with these data sets in both the change trend of variables and their magnitude values, and the new scheme, together with the algorithm, is more efficient and saves more computer time.展开更多
The seasonal frozen soil on the Qinghai-Tibet Plateau has strong response to climate change, and its freezing-thawing process also affects East Asia climate. In this paper, the freezing soil maximum depth of 46 statio...The seasonal frozen soil on the Qinghai-Tibet Plateau has strong response to climate change, and its freezing-thawing process also affects East Asia climate. In this paper, the freezing soil maximum depth of 46 stations covering 1961-1999 on the plateau is analyzed by rotated experience orthogonal function (REOF). The results show that there are four main frozen anomaly regions on the plateau, i.e., the northeastern, southeastern and southern parts of the plateau and Qaidam Basin. The freezing soil depths of the annual anomaly regions in the above representative stations show that there are different changing trends. The main trend, except for the Qaidam Basin, has been decreasing since the 1980s, a sign of the climate warming. Compared with the 1980s, on the average, the maximum soil depth decreased by about 0.02 m, 0.05 m and 0.14 m in the northeastern, southeastern and southern parts of the plateau, but increased by about 0.57 m in the Qaidam Basin during the 1990s. It means there are different responses to climate system in the above areas. The spectrum analysis reveals different change cycles: in higher frequency there is an about 2-year long cycle in Qaidam Basin and southern part of the plateau in the four representative areas whereas in lower frequency there is an about 14-year long cycle in all the four representative areas due to the combined influence of different soil textures and solutes in four areas.展开更多
Frozen soils cover about 40%of the land surface on the earth and are responsible for the global energy balances affecting the climate.Measurement of the thermal properties of frozen soils during phase transition is im...Frozen soils cover about 40%of the land surface on the earth and are responsible for the global energy balances affecting the climate.Measurement of the thermal properties of frozen soils during phase transition is important for analyzing the thermal transport process.Due to the involvement of phase transition,the thermal properties of frozen soils are rather complex.This paper introduces the uses of a multifunctional instrument that integrates time domain re fl ectometry(TDR)sensor and thermal pulse technology(TPT)to measure the thermal properties of soil during phase transition.With this method,the extent of phase transition(freezing/thawing)was measured with the TDR module;and the corresponding thermal properties were measured with the TPT module.Therefore,the variation of thermal properties with the extent of freezing/thawing can be obtained.Wet soils were used to demonstrate the performance of this measurement method.The performance of individual modules was fi rst validated with designed experiments.The new sensor was then used to monitor the properties of soils during freezing e thawing process,from which the freezing/thawing degree and thermal properties were simultaneously measured.The results are consistent with documented trends of thermal properties variations.展开更多
The average temperature of frozen soil wall is an essential parameter in the process of design,construction,and safety management of artificial ground freezing engineering.It is the basis of calculating frozen soil...The average temperature of frozen soil wall is an essential parameter in the process of design,construction,and safety management of artificial ground freezing engineering.It is the basis of calculating frozen soil's mechanical parameters,further prediction of bearing capacity and,ultimately,safety evaluation of the frozen soil wall.Regarding the average temperature of single-row-piped frozen soil wall,this paper summarizes several current calculation methods and their shortcomings.Furthermore,on the basis of Bakholdin's analytical solution for the temperature field under straight single-row-piped freezing,two new calculation models,namely,the equivalent trapezoid model and the equivalent triangle model,are proposed.These two approaches are used to calculate the average temperature of a certain cross section which indicates the condition of the whole frozen soil wall.Considering the possible parameter range according to the freezing pipe layout that might be applied in actual construction,this paper compares the average temperatures of frozen soil walls obtained by the equivalent trapezoid method and the equivalent triangle method with that obtained by numerical integration of Bakholdin's analytical solution.The results show that the discrepancies are extremely small and these two new approaches are better than currently prevailing methods.However,the equivalent triangle method boasts higher accuracy and a simpler formula compared with the equivalent trapezoid method.展开更多
基金supported by the National Natural Science Foundation of China (No. 41471062, No. 41971085, No. 41971086)。
文摘The warm and ice-rich frozen soil is characterized by high unfrozen water content, low shear strength and large compressibility, which is unreliable to meet the stability requirements of engineering infrastructures and foundations in permafrost regions. In this study, a novel approach for stabilizing the warm and ice-rich frozen soil with sulphoaluminate cement was proposed based on chemical stabilization. The mechanical behaviors of the stabilized soil, such as strength and stress-strain relationship, were investigated through a series of triaxial compression tests conducted at -1.0℃, and the mechanism of strength variations of the stabilized soil was also explained based on scanning electron microscope test. The investigations indicated that the strength of stabilized soil to resist failure has been improved, and the linear Mohr-Coulomb criteria can accurately reflect the shear strength of stabilized soil under various applied confining pressure. The increase in both curing age and cement mixing ratio were favorable to the growth of cohesion and internal friction angle. More importantly, the strength improvement mechanism of the stabilized soil is attributed to the formation of structural skeleton and the generation of cementitious hydration products within itself. Therefore, the investigations conducted in this study provide valuable references for chemical stabilization of warm and ice-rich frozen ground, thereby providing a basis for in-situ ground improvement for reinforcing warm and ice-rich permafrost foundations by soil-cement column installation.
基金supported by the National Natural Science Foundation of China (No. 41471062, No. 41971085, No. 41971086)。
文摘Affected by climate warming and anthropogenic disturbances, the thermo-mechanical stability of warm and ice-rich frozen ground along the Qinghai-Tibet engineering corridor(QTEC) is continuously decreased, which may delay the construction of major projects in the future. In this study, based on chemical stabilization of warm and icerich frozen ground, the soil-cement column(SCC) for ground improvement was recommended to reinforce the foundations in warm and ice-rich permafrost regions. To explore the validity of countermeasures mentioned above, both the original foundation and the composite foundation consisting of SCC with soil temperature of -1.0℃ were prepared in the laboratory, and then the plate loading tests were carried out. The laboratory investigations indicated that the bearing capacity of composite foundation consisting of SCC was higher than that of original foundation, and the total deformation of original foundation was greater than that of composite foundation, meaning that overall stability of foundation with warm and ice-rich frozen soil can be improved by SCC installation. Meanwhile, a numerical model considering the interface interaction between frozen soil and SCC was established for interpretating the bearing mechanism of composite foundation. The numerical investigations revealed that the SCC within composite foundation was responsible for the more applied load, and the applied load can be delivered to deeper zone in depth due to the SCC installation, which was favorable for improving the bearing characteristic of composite foundation. The investigations provide the valuable guideline for the choice of engineering supporting techniques to major projects within the QTEC.
基金supported by the National Natural Science Foundation of China(Grant No.41971086)Natural Science Foundation of Shanxi Province(Grant No.2023-JC-QN-0626,2022JQ-467).
文摘In permafrost regions with warm frozen soil,subgrade thaw-collapse phenomenon commonly occurs,facing thaw collapse problems of the existed frozen soil subgrade,thus it is difficult to use traditional methods such as active cooling and passive protection technology to stabilize the existed warm frozen soil subgrade.This study derives a novel stabilizer method,a long-short(L-S)cement-mixed batter pile composite foundation to stabilize the existed warm frozen soil subgrade.To solve the thawcollapse problems in warm frozen soil subgrade,high water content and large compressibility characteristics were compared between soft soil and warm frozen soils.Theoretical analysis of heat conduction and numerical simulation of finite element model were used to study the freeze–thaw process and evaluate the stabilized effects of the L-S cement-mixed batter piles on the warm frozen soil foundation of the Qinghai-Tibet Highway.Furthermore,the thaw process and mechanical properties of foundation and piles were analyzed by introducing the hydration heat factor in the thermodynamic control equation.The results indicate that the thawing displacement of the existed warm frozen soil subgrade was reduced owing to the“support”and“grasp”effects of the L-S cement-mixed batter piles on the surrounding soil.The composite ground formed by strengthening the warm frozen ground with batter piles could considerably improve the bearing capacity of the existed warm frozen ground,effectively restrain the deformation of the upper embankment,and improve the strength of the ground.The analysis can provide method for the construction design of cement mixing batter pile foundation in cold regions.
文摘A rate-dependent constitutive model for saturated frozen soil is vital in frozen soil mechanics,especially when simultaneously describing the nonlinearity,dilatancy and strain-softening characteristics.The distribution of the non-uniform strain rate of saturated frozen soil at the meso-scale due to the local icecementation breakage is described by a newly binary-medium-based homogenization equation.Based on the field-equation-based approach of the meso-mechanics theory,the interaction expression of the strain rate at macro-and meso-scale is derived,which can give the strain rate concentration tensor at different crushed degrees.With the thermodynamics and empirical assumption,a breakage ratio in the rate-dependent form is determined.This overcomes the limitations of the existing binary-medium-based models that are difficult to simulate rate-dependent mechanical response.Based on these assumptions,a newly binary-medium-based rate-dependent model is proposed considering both the ice bond breakage and material composition characteristics of saturated frozen soil.The proposed constitutive model has been validated by the test results on frozen soils with different temperatures and strain rates.
基金This work was supported by the National Key R&D Program of(Grant No.2016YFA0602302).
文摘The acquisition of spatial-temporal information of frozen soil is fundamental for the study of frozen soil dynamics and its feedback to climate change in cold regions.With advancement of remote sensing and better understanding of frozen soil dynamics,discrimination of freeze and thaw status of surface soil based on passive microwave remote sensing and numerical simulation of frozen soil processes under water and heat transfer principles provides valuable means for regional and global frozen soil dynamic monitoring and systematic spatial-temporal responses to global change.However,as an important data source of frozen soil processes,remotely sensed information has not yet been fully utilized in the numerical simulation of frozen soil processes.Although great progress has been made in remote sensing and frozen soil physics,yet few frozen soil research has been done on the application of remotely sensed information in association with the numerical model for frozen soil process studies.In the present study,a distributed numerical model for frozen soil dynamic studies based on coupled water-heat transferring theory in association with remotely sensed frozen soil datasets was developed.In order to reduce the uncertainty of the simulation,the remotely sensed frozen soil information was used to monitor and modify relevant parameters in the process of model simulation.The remotely sensed information and numerically simulated spatial-temporal frozen soil processes were validated by in-situ field observations in cold regions near the town of Naqu on the East-Central Tibetan Plateau.The results suggest that the overall accuracy of the algorithm for discriminating freeze and thaw status of surface soil based on passive microwave remote sensing was more than 95%.These results provided an accurate initial freeze and thaw status of surface soil for coupling and calibrating the numerical model of this study.The numerically simulated frozen soil processes demonstrated good performance of the distributed numerical model based on the coupled water-heat transferring theory.The relatively larger uncertainties of the numerical model were found in alternating periods between freezing and thawing of surface soil.The average accuracy increased by about 5%after integrating remotely sensed information on the surface soil.The simulation accuracy was significantly improved,especially in transition periods between freezing and thawing of the surface soil.
基金This work was supported by the National Natural Science Foundation of China(No.41472253)the Key project of Natural Science Foundation of Tianjin City(No.6JCZDJC39000).
文摘The correct determination of thermal parameters,such as thermal conductivity and specific heat of soil during freezing,is the most important and basic problem for the construction of an appropriate freezing method.In this study,a calculation model of three stages of soil temperature was established.At the unfrozen and frozen stages,the specific temperatures of dry soil,water,and ice are known.According to the principle of superposition,a calculation model of unfrozen and frozen soils can be established.Informed by a laboratory experiment,the latent heat of the adjacent zone was calculated for the freezing stage based on different water contents in the temperature section.Both the latent and specific heat of water,ice,and particles were calculated via superposition of the weight percentage content.A calculation model of the specific heat of the freezing stage was built,which provides both guidance and theoretical basis for the calculation of the specific heat of frozen soil.
基金the Natural Key Research and Development Program of China(Grant No.2017YFC0405103).
文摘Numerical simulation is known as an effective method for mechanical properties during frozen soil excavation.In order to reveal the development of cutting force,effective stress and cutting fragments in frozen silt during the cutting process,we introduce an explicit finite element program LS-DYNA to establish a two-dimensional numerical model of the frozen soil cut.We also use the Holmquist-Johnson-Cook(HJC)damage constitutive model for simulating the variation of soil mechanical properties according to the strong dependence between the cutting tool and frozen silt during the process with different cutting depths,angles and velocities.Meanwhile,a series of experimental results are acquired of frozen silt cutting to prove the application of the HJC model during simulation of cutting force variations.The result shows that the cutting force and fragment size are strongly influenced by cutting depths and cutting velocities increased,and the maximum effective stress at points where the tool contacts frozen soil during the cutting process.In addition,when the cutting angle is 52°,the cutting force is the smallest,and the cutting angle is optimum.Thus,the prediction of frozen soil mechanical properties on the cutting process by this model is conducive to selecting machinery equipment in the field.
基金Supported by the Application Technology Research and Development Plan Project of Heilongjiang Province(GY2014ZB0011)the 13th Five-year National Key R&D Program(2016YFD0300610)
文摘In view of the cumbersome and often untimely process of manual collection and observation of frozen soil data parameters,and the damage caused to dams by frost heaving of frozen soil,a remote monitoring and an early warning model for frozen soil in dam areas was presented.The Pt100 temperature sensors and JM seam gauges were used as measurement tools in the system.The sensor layout was designed,based on the actual situation in the monitoring area.A 4G network was used for wireless transmission to monitor frozen soil data in real time.BP neural network was used to predict the parameters of frozen soil.After analysis,four factors including the average temperature of frozen soil,the type of frozen soil,the artificial upper limit of frozen soil and the building construction time were selected to establish an early warning model using fuzzy reasoning.The experimental results showed that the early warning model could reflect the influence on dam buildings of frost heaving and sinking of frozen soil,and provided technical support for predicting the hazard level.
基金supported by the National Natural Science Foundation of China(Nos.U1703244 and 41672310)the National Natural Science Foundation of China(No.41801038)+6 种基金the State Key Laboratory for Geo Mechanics and Deep Underground Engineering,the China University of Mining and Technology(SKLGDUEK1904)the Strategic Priority Research Program of the Chinese Academy of Sciences(Grant No.XDA2003020102)the Major Program of Bureau of International Cooperation,the Chinese Academy of Sciences(131B62KYSB20170012)the National Key Research and Development Program(2017YFC0405101)the Research Project of the State Key Laboratory of Frozen Soils Engineering(Grant No.SKLFSE-ZY-16)the Science and Technology Major Project of Gansu Province(143GKDA007)the Science and Technology Planning Project of Gansu Province(No.18JR3RA376)
文摘A series of directional shear tests on remolded frozen soil was carried out at 10°C by using a hollow cylinder apparatus to study failure criterion under a directional shear-stress path.Directional shear tests were conducted at five shear rates(10,20,30,40,and 50 kPa/min)and five intermediate principal stress coefficients(b=0,0.25,0.5,0.75,and 1),with the mean principal stress(p=4.5 MPa)kept constant.The results show that the torsional strength and the generalized strength both increase with the increase of the shear rates.According to the failure modes of frozen soil under different shear rates,the specimens present obvious plastic failure and shear band;and the torsional shear component dominates the failure modes of hollow cylindrical specimens.A shear rate of 30 kPa/min is chosen as the loading rate in the directional shear tests of frozen soil.The shape of the failure curve in theπplane is dependent on the directional anglesαof the major prin cipal stress.It is reasonable to use the strain-hardening curves to define the deviatoric stress value atγg=15%(generalized shear strain)as the failure criterion of frozen soil under a directional shear-stress path.
基金The research is supported by the Natural Science Foundation of Anhui Province(Grant No.1908085QE242)the Fundamental Research Funds for the Central Universities(Grant No.JZ2021HGTB0097)the Natural Science Foundation of China(NSFC)(Grant No.51908175).The financial support is gratefully acknowledged.
文摘Soil slope stability in seasonally frozen regions is a challenging problem for geotechnical engineers.The freezethaw process of soil slope caused by the temperature fluctuation increases the difficulty in predicting the slope stability because the soil property is influenced by the freeze-thaw cycle.In addition,the frozen soil,which has ice crystal,ice lens and experienced freeze-thaw process,could present stronger heterogeneity.Previous research has not investigated the combined effect of soil heterogeneity and freeze-thaw cycle.This paper studies the influence of soil heterogeneity on the stability of frozen soil slope under freeze-thaw cycles.The local average subdivision(LAS)is utilized to model the soil heterogeneity.A typical slope geometry has been chosen and analysed as an illustrative example and the strength reduction method is used to calculate the factor of safety(FOS)of slope.It has been found that when the temperature is steady,the FOS of the frozen soil slope is influenced by the spatial variability of the thermal conductivity,but the influence is not significant.When the standard deviation and the SOF of the thermal conductivity increase,the mean of the FOS is equal to the FOS of the homogeneous case and the standard deviation of the FOS also increases.After the frozen soil goes through freeze-thaw process,the FOS of the frozen soil slope decreases due to the reduction in the cohesion and the internal friction angle caused by the freeze-thaw cycles.Furthermore,the decreasing ratio of the FOS becomes more scattered after the 5th freeze-thaw cycle compared to that of the FOS after the 1st freeze-thaw cycle.The larger variability of the FOS could induce inaccuracy in the prediction of the frozen soil slope stability.
基金supported by the National Natural Science Foundation of China(Grant No.51978039)the Fundamental Research Funds for the Central Universities(Grant No.2021YJS115)。
文摘The vibration of underground or buried piping during construction and long-term operation causes secondary disasters,and becomes more complex when tubes are buried in cold regions.The interface between saturated frozen soil and lining is regarded as a thin spring-like layer whose thickness could be negligible.In this paper,the dynamic response of saturated frozen soil is studied in frequency domain by using the Helmholtz composition and Fourier transform to obtain analytical solutions of the radial and axial displacement,as well as expressions of the stiffness coefficient(Kr)and damping coefficient(Cr)of the spring-like interface.Numerical results indicate that Krand Crare related to physical properties of the lining and its surrounding soil,and the coefficients of the springlike model could be changed by adjusting lining parameters to improve structure stability under the same load conditions.Also,the viscoelastic contact surface of the spring-like model is considered to have less effect on the surrounding soil than that when the lining has complete contact with the soil under load.The degree of soil freezing significantly affects the axial and radial displacement of the soil when the interface between lining and unsaturated frozen soil is taken into consideration.
基金the National Natural Science Foundation of China(Grant No.41877251)Research and Practice Project of Higher Education Teaching Reform in Henan Province,China(Grant No.2019SJGLX463)。
文摘The freezing of soil containing a liquid is a complex transient heat conduction problem involving phase change and release or absorption of latent heat.Existing efforts have essentially focused on theoretical research and numerical simulations.In the present study,the problem is approached from an experimental point of view using the so-called“freezing model test”method.In particular,in order to establish a precise relationship between the model and the prototype,a temperature similarity criterion is derived using the condition of an equal number of Kosovitch.Similarity is also established with respect to other aspects.A similarity criterion for the water field is determined on the basis of relevant partial differential equations.Analogous criteria for the stress field and load are derived using an elastic model.The validity of this approach is experimentally verified.The research results provide a practical and reasonable method for calculating the parameters for preparing model soils.They also constitute a theoretical basis and a technical support for the design and implementation of a water-heat-force similarity coupled framework.
文摘Seasonally frozen soil in alpine and subalpine zones in the mountains of Qinghai-Tibetan Plateau is particularly sensitive to global climate change. Therefore, a better understanding of the thermal properties of frozen soil is crucial for predicting the responses of frozen soils to soil warming. In this study, thermal properties of frozen soil with different moisture contents under subzero temperature (0°C - 20°C) in an alpine forest in western Sichuan were analyzed by KD<sub>2</sub> Pro in its cooling and heating processes, respectively. Our results reveal that the soil apparent volumetric specific heat capacity (C<sub>v</sub>) and apparent thermal conductivity (K) under the same water content show similar response patterns to changing temperature lower than -2°C in both heating and cooling processes. Moreover, ice content of frozen soils can be well predicted by Logistic model in cooling and heating processes. The C<sub>v</sub> and K tend to increase along with increasing soil moisture contents. Remarkably, asymptotic characters of the value of C<sub>v</sub> and K are at the vicinity of the initial temperature of phase transitions, indicating that both C<sub>v</sub> and K are particularly sensitive to changing soil temperature at the range of -2°C to 0°C. Therefore, the widely distributed frozen soil layers with temperature above -2°C in alpine and subalpine zones over Qinghai-Tibetan Plateau are susceptible to the observed climate warming during cold season.
基金supported by the National Natural Science Foundation of China (No.11172251)the Open Fund of State Key Laboratory of Frozen Soil Engineering (No.SKLFSE201001)the Fundamental Research Funds for the Central Universities (No.SWJTU09CX069)
文摘The split Hopkinson pressure bar (SHPB) method is used to investigate the dynamic behavior of the artificial frozen soil under the nearly uniaxial strain and uniaxial stress conditions. The tests are conducted at the temperatures of 3℃, 8℃, 13℃, 17℃, 23℃, and 28℃ and with the strain rates from 900 s-1 to 1 500 s-1 . The nearly uniaxial stress-strain curves exhibit an elastic-plastic behavior, whereas the uniaxial stress-strain curves show a brittle behavior. The compressive strength of the frozen soil exhibits the positive strain rate and negative temperature sensitivity, and the final strain of the frozen soil shows the positive strain rate sensitivity. The strength of the frozen soil under the nearly uniaxial strain is greater than that under the uniaxial stress. After the negative confinement tests, the specimens are compressed, and the visible cracks are not observed. However, the specimens are catastrophically damaged after the uniaxial SHPB tests. A phenomenological model with the thermal sensitivity is established to describe the dynamic behavior of the confined frozen soil.
基金supported by the National Natural Science Foundation of China (Grant Nos. 41471062 and 41401087)the State Key Laboratory of Frozen Soil Engineering (Grant No.SKLFSE-ZT-35)
文摘Warm and ice-rich frozen soil(WIRFS) exhibits lower shear strength due to the weak binding forces between soil particles and ice crystals. To enhance the strength of WIRFS, frozen soil was treated separately with Portland, Phosphate, Sulphoaluminate, Portland-Phosphate and PortlandSulphoaluminate cements. After the samples were cured under -1.0°C for 7 days, the microscopic pore distribution characteristics and the macro-mechanical properties of WIRFS were investigated using mercury intrusion porosimetry(MIP), scanning electron microscopy(SEM) and unconfined compressive strength(UCS) tests. To quantitatively analyze the laws of pore-size transformation and the variation of Hausdorff volumetric fractal dimensions for pre-and post-treated WIRFS, the CURVEEXTRACT and Image-Pro Plus(IPP) image analysis system has been developed for analysing SEM images of the soil samples. Statistics of the pore-area dimension and pore-volume dimension were calculated. The results reveal that the cement-based treatment of WIRFS can improve the cementation fill of soil pores and the bond forces between soil particles. There is an evident correlation between the microstructure characteristics and the mechanical properties of the treated WIRFS. As the fractal dimensions of pore-area decrease, the unconfined compressive strength of cement-treated WIRFS increases significantly. In contrast, as the fractal dimensions of pore-volume increases, the unconfined compressive strength decreases remarkably.
基金supported by the National Natural Science Foundation of China (Grant No. 41171064)the National Basic Research Program of China (973 Program Grant No. 2012CB026104)
文摘Using newly developed dynamic shearing devices, the dynamic shearing strength of frozen soil-concrete interface was studied experimentally. By placing concrete blocks in the lower half of the shear box and frozen soil sample in the upper part, a series of dynamic shear tests on their interfaces were carried out. The obtained results are summarized and the main influencing factors are revealed.
基金the National Natural Science Foun-dation of China under Grant Nos. 40575043 and 40605024as well as 40730952the National Basic Research Program of China under Grant No. 2009CB421405The Innovation Project of the Chinese Academy of Sci-ences (Grant No. KZCX2-YW-220)
文摘In almost all frozen soil models used currently, three variables of temperature, ice content and moisture content are used as prognostic variables and the rate term, accounting for the contribution of the phase change between water and ice, is shown explicitly in both the energy and mass balance equations. The models must be solved by a numerical method with an iterative process, and the rate term of the phase change needs to be pre-estimated at the beginning in each iteration step. Since the rate term of the phase change in the energy equation is closely related to the release or absorption of the great amount of fusion heat, a small error in the rate term estimation will introduce greater error in the energy balance, which will amplify the error in the temperature calculation and in turn, cause problems for the numerical solution convergence. In this work, in order to ?rst reduce the trouble, the methodology of the variable transformation is applied to a simpli?ed frozen soil model used currently, which leads to new frozen soil scheme used in this work. In the new scheme, the enthalpy and the total water equivalent are used as predictive variables in the governing equations to replace temperature, volumetric soil moisture and ice content used in many current models. By doing so, the rate terms of the phase change are not shown explicitly in both the mass and energy equations and its pre-estimation is avoided. Secondly, in order to solve this new scheme more functionally, the development of the numerical scheme to the new scheme is described and a numerical algorithm appropriate to the numerical scheme is developed. In order to evaluate the new scheme of the frozen soil model and its relevant algorithm, a series of model evaluations are conducted by comparing numerical results from the new model scheme with three observational data sets. The comparisons show that the results from the model are in good agreement with these data sets in both the change trend of variables and their magnitude values, and the new scheme, together with the algorithm, is more efficient and saves more computer time.
基金Key project of CAS, No.KZCX1-10-07 Key project of Cold and Arid Regions Environmental and Engineering Research Institute, CAS, No.CX210097 NSFC No.49805006.
文摘The seasonal frozen soil on the Qinghai-Tibet Plateau has strong response to climate change, and its freezing-thawing process also affects East Asia climate. In this paper, the freezing soil maximum depth of 46 stations covering 1961-1999 on the plateau is analyzed by rotated experience orthogonal function (REOF). The results show that there are four main frozen anomaly regions on the plateau, i.e., the northeastern, southeastern and southern parts of the plateau and Qaidam Basin. The freezing soil depths of the annual anomaly regions in the above representative stations show that there are different changing trends. The main trend, except for the Qaidam Basin, has been decreasing since the 1980s, a sign of the climate warming. Compared with the 1980s, on the average, the maximum soil depth decreased by about 0.02 m, 0.05 m and 0.14 m in the northeastern, southeastern and southern parts of the plateau, but increased by about 0.57 m in the Qaidam Basin during the 1990s. It means there are different responses to climate system in the above areas. The spectrum analysis reveals different change cycles: in higher frequency there is an about 2-year long cycle in Qaidam Basin and southern part of the plateau in the four representative areas whereas in lower frequency there is an about 14-year long cycle in all the four representative areas due to the combined influence of different soil textures and solutes in four areas.
文摘Frozen soils cover about 40%of the land surface on the earth and are responsible for the global energy balances affecting the climate.Measurement of the thermal properties of frozen soils during phase transition is important for analyzing the thermal transport process.Due to the involvement of phase transition,the thermal properties of frozen soils are rather complex.This paper introduces the uses of a multifunctional instrument that integrates time domain re fl ectometry(TDR)sensor and thermal pulse technology(TPT)to measure the thermal properties of soil during phase transition.With this method,the extent of phase transition(freezing/thawing)was measured with the TDR module;and the corresponding thermal properties were measured with the TPT module.Therefore,the variation of thermal properties with the extent of freezing/thawing can be obtained.Wet soils were used to demonstrate the performance of this measurement method.The performance of individual modules was fi rst validated with designed experiments.The new sensor was then used to monitor the properties of soils during freezing e thawing process,from which the freezing/thawing degree and thermal properties were simultaneously measured.The results are consistent with documented trends of thermal properties variations.
基金supported by the National Natural Science Foundation of China (No. 50578120)the National High Technology Research and Development Program of China (863 Program) (No. 2006AA11Z118)
文摘The average temperature of frozen soil wall is an essential parameter in the process of design,construction,and safety management of artificial ground freezing engineering.It is the basis of calculating frozen soil's mechanical parameters,further prediction of bearing capacity and,ultimately,safety evaluation of the frozen soil wall.Regarding the average temperature of single-row-piped frozen soil wall,this paper summarizes several current calculation methods and their shortcomings.Furthermore,on the basis of Bakholdin's analytical solution for the temperature field under straight single-row-piped freezing,two new calculation models,namely,the equivalent trapezoid model and the equivalent triangle model,are proposed.These two approaches are used to calculate the average temperature of a certain cross section which indicates the condition of the whole frozen soil wall.Considering the possible parameter range according to the freezing pipe layout that might be applied in actual construction,this paper compares the average temperatures of frozen soil walls obtained by the equivalent trapezoid method and the equivalent triangle method with that obtained by numerical integration of Bakholdin's analytical solution.The results show that the discrepancies are extremely small and these two new approaches are better than currently prevailing methods.However,the equivalent triangle method boasts higher accuracy and a simpler formula compared with the equivalent trapezoid method.
