Objective

The construction of pile foundation projects in cold regions requires quantitative analyses of displacement and stress of surrounding soil

which can be realized by employing cavity expansion methods. The classical cavity expansion methods fail to take consideration of uneven temperature distribution

temperature change and unique mechanical behavior of frozen soil. Thus

this paper proposed a thermo-elasto-plastic model based on the strength criterion with a parabolic curve (also called parabolic strength criterion) applicable to frozen soil

giving expressions for displacement and stres

s.

Methods

Firstly

the geometry and boundary conditions were determined

and several assumptions of the proposed model were introduced. Dirichlet boundary condition (a given fixed surface temperature) was considered in addressing heat diffusion problem. Secondly

the governing equations of thermo-elastic cavity expansion model were introduced and combined together

by which the expressions of displacement and stress in elastic state can be derived explicitly. With the increase of internal radial pressure

the state of surrounding soil will gradually transit from elastic state to elasto-plastic state

by which the surrounding soil can be divided into elastic zone and plastic zone with different governing equations. Thirdly

in order to consider the nonlinear strength of frozen soil

the parabolic strength criterion was introduced

i.e.

the strength first increases and then decreases as the mean stress increases. The stress distribution of plastic zone can be derived by combining equilibrium equations of continuum with parabolic strength criterion of frozen soil. It is obviously impossible to derive expression for displacement via ordinary routine in plasticity since the parabolic strength criterion was based on triaxial experiment. As a result

the proposed model assumes that the compressive equations remain unchanged after the transition from elastic to plastic state. Substituting the compressive equations into kinematics of continuum mechanics

the expression for the displacement of internal wall surface can be derived. Finally

an example of pile foundation construction was analyzed based on the proposed model and the calculation result were compared with that of precedented model and numerical method.

Results and Discussions

Comparing the procedure of modeling with that of classical cavity expansion model

both stress and displacement were interconnected with temperature distribution. A logarithm temperature distribution was firstly obtained by solving heat diffusion equation

which

was also used in the derivation of elastic stress and displacement. As for the stress and displacement in elastic state

the effect of temperature was easily seen from the term including thermal expansion coefficient which is usually small for solids such as ice

rock and soil. A significant discovery is that when compressive stress was set positive

the radial stress was always maximum principal stress and the circumferential stress was always minimum principal stress. This discovery facilitated the determination of plastic zone and the substitution of principal stress into parabolic strength criterion. In plastic zone

the effect of temperature was far more sophisticated because the variation of temperature not only changed the parameters of parabolic strength criterion

but also affected the subsequent calculation of stress and displacement. The exercise of parabolic strength criterion makes it difficult to obtain explicit expression for stress and displacement in plastic zone

while quantitative calculation is possible via numerical analysis. The quantitative result of the example demonstrated some unique features of the proposed model. For example

with the increase of internal radial pressure

the plastic boundary gradually moved outward

at 1 m when the pressure was 1690 kPa

at 2 m when the pressure was 6306 kPa. It is also noted that the quadratic approximation to the coordinate of plastic boundary worked well. Meanwhile

the circumferential stress on the boundary surface of plastic zone was found to be discontinuous

which can be attributed to the stress redistribution of frozen soil. The displacement of the internal wall surface was calculated subsequently

which was 83.5 mm when the pressure was 1690 kPa

and was 170.0 mm when the pressure was 5000 kPa. The property of compressive hardening can be easily observed from the stress-displacement curve. Comparing the result with that of precedented model based on Tresca strength criterion and of numerical method based on PFEM

the stress-displacement rela

tion was acceptable from the perspective of geotechnics. The sensitivity of parameters was further analyzed and mainly concentrated on the effect of temperature and coefficient of plasticity-growing. The internal temperature groups of -5 ℃

-10 ℃

-15 ℃ were set respectively

resulting in smaller plastic zone (showing higher strength of frozen soil) and lower circumferential stress drop at the plastic boundary. And the coefficient of plasticity-growing can be used to further modify the results.

Conclusions

The proposed model is based on the strength criterion with a parabolic curve applicable to frozen soil

taking temperature variation into consideration

using thermoelastic constitutive equation which treats temperature as a variable to derive the expressions for the stress and displacement of soil. This model is able to incorporate more complex conditions in the construction and is verified to be effective and rational from the perspective of geotechnics

which is instructive for the in-situ soil testing

design and construction of pile foundation in frozen soil regions.