Calculation of Base Settlement on Permafrost Ground According to SNIP 2.02.0488
The article is devoted to methods of base settlement on permafrost calculation with detailed description according to SNIP 2.02.0488.
INTRODUCTION
The calculation of bases in permafrost regions is quite a complicated and specialized process, heavily influenced by thermal field and thawing phenomena. Thermal field causes permafrost thawing and decreases its load bearing capacity while increasing ground base deformation, which is usually evidenced in the form of base settlement. In fact, to determine the deformations during building maintenance, it is necessary to solve for the stressstrain state of the base, which is described by differential equations of equilibrium and the laws of elasticplastic deformation.
Calculations can be based on numerical methods which solve the nonlinear differential equations in 3D and do not require any assumptions to overly simplify the stressstrain state model. The main disadvantage of using numerical methods to solve a stressstrain problem is the need for big computational resources. Therefore, modern PCs are not capable of solving stressstrain problems of large spatial scale.
There are, however, analytical methods for simulating base deformation in some particular cases. These methods cannot solve common problems, but still find practical implementation in some circumstances.
SNIP 2.02.0488
A set of norms regulating building practices, which also sets down guidelines (with links to other technical documents) for the calculation of bases and foundations.
According to SNIP 2.02.0488, there are two basic principles involving permafrost ground bases:
Principle I  Principle II 
permafrost grounds in the frozen state must remain thus during construction and throughout the operating period  permafrost bases must either be fully or partially thawed (with warming on the computed depth before construction begins or during building maintenance) 
The first principle therefore implies the preservation of the initial state of the ground during use, while the second allows for changes in ground base states. Both principles involve deformation of the base due to the load of the structure. Thawing grounds can lead to significant base deformation, which can compromise structural integrity. In such cases, steps must be taken to reduce base deformation and / or provide a means for buildings to adjust to the nonuniform deformation of their base. The decision to employ certain measures should be based on the calculated values of base settlement, which, as previously mentioned, is the main indicator of base deformation.
Calculation method for base settlement on permafrost, in compliance with the SNIP 2.02.0488 "Bases and Foundations on Permafrost Grounds".
Settlement is the difference in the height of a ground surface point between the initial time and the time for which the prediction is made. If we set the initial time and prediction time as t_{0} and t respectively, and describe height of the surface as a function depending on the time h(τ), then the settlement s(t_{0}, t)=h(t)  h(t_{0}).
Initial time t_{0} represents the time, when the initial model of the problem was created (initial conditions was measured). In fact t_{0} can be considered not so much a time, but as a set of physical parameters of the initial model or the initial conditions.
Prediction time t is the time for which prediction is made (t may be called the simulated time). As with the parameter t_{0}, t can be regarded as a set of physical parameters (the predicted conditions) needed to calculate settlement.
GROUND MODEL
Calculation of settlement according to SNIP 2.02.0488 is applied to a point on the ground surface (the design point).
For further calculation, a computational model of the ground is built, assuming homogenous ground layers (i.e., the ground layer is formed by one material) and uniform layer temperature and ice content. The ground model that corresponds to this standard implies n≥1 homogeneous layers in the ground, greatly simplifying the model because the layers are not threedimensional, but simply defined in the vertical plane.
Let’s consider a vertical line intersecting the design points. The vertical intersects the ground layers at certain heights (Fig. 1).
Fig. 1: Ground model for settlement calculation
Fig. 1 shows two design points (B and C). There are 4 layers L_{1},L_{2},...,L_{n}, in the ground model and the bottom is bounded by a fixed base (assuming that the fixed base is not affected by settlement). The vertical lines passing through the design points intersect all ground layers L_{1},L_{2},...,L_{n} (n=4 in our example), at the heights h_{1},h_{2},…,h_{n} and the height of the fixed base z_{0} (instead of z_{0} the height of a design point can be used). Please note that the L_{i} – a model of the ith layer – may include a set of parameters (detailed below).
The model is onedimensional and does not include ground settlement due to horizontal deformation.
MODEL OF THE LAYER
Each homogeneous layer can be defined with a material parameter M_{i} a thickness h_{i}, a temperature T_{i}, ice content I_{i} and pressures P_{i} at the top and P_{i} at the deeper layers.
The ice content of the layer is the ratio of frozen ground mass to the total weight of the ground.
The material M_{i}, is assumed constant. The temperature T_{i}, ice content I_{i} and height (layer thickness) h_{i} parameters are closely related: ice content depends on the temperature, while thickness depends on ice content. In practice, the parameters T_{i}, I_{i} and h_{i} change over the time interval. The parameter P_{i} does not depend on the ith layer. P_{i}=P_{i}pressure caused by the weight of ith layer. The mass of the ith layer only varies with water migration, if this phenomenon is not taken into account, the mass of the ith layer is constant.
The layer model can thus be defined:
L_{i}=(M_{i},T_{i},I_{i},h_{i},P_{i},P_{i})
This model describes the settlement at a particular time. The initial layer model (at the initial time) can be described as follows:
L_{i}^{0}=(M_{i}^{0},T_{i}^{0},I_{i}^{0},h_{i}^{0},P_{i}^{0},P_{i}^{0}) 
(7.19) 
GENERAL FORMULA FOR SETTLEMENT CALCULATION
According to SNIP 2.02.0488, the equation for settlement is:
s = s_{th} + s_{p} 
where s_{th} is the base settlement component due to the action of thawing ground weight; and
s_{p} is the base settlement component caused by the additional pressure of structure on the ground.
Calculation of settlement due to layer weight is:

(7.20) 
n is the number of ground layers in the calculation;
A_{th}_{,i} and m_{th}_{,i} are the thawing coefficient (dimensionless) and the compressibility factor (kPa^{1}) of ith layer of thawing ground (experimental data), respectively.
σ_{zg}_{,i} is the vertical stress of ground weight at the middle of the ith layer (kPa), determined by allowing for the buoyancy property of water. The middle of ith layer is on the depth z_{i} from the leveling marks;
h_{i} is the ith layer thickness of thawing ground, in meters.
Note: The SNIP Norms account for either frozen or thawed (but never partially thawed ground). In formula (7.20), the summing routines only consider the layers that have passed out of frozen state into the thawed one.
For the convenience of calculation, the following notations are changed:
The material parameters of the ith layer A_{th}_{,}_{i} and m_{th}_{,}_{i } are simplified to: A_{i}:=A_{th,i} and m_{i}:=m_{th,i}, M_{i}:=(A_{i},m_{i}).
The initial thickness of ith layer h_{i} is represented with h_{i}^{0}, to highlight that the measurements were done at the initial time.
Instead of σ_{zg}_{,i} (stress in the middle of the ith layer), we will use the pressure at the top of the ith layer P_{i}, and pressure below ith layer P_{i}:
Rewriting the formula for s_{th} with the new notation:

(7.20a) 
Formula (7.20) and (7.20a) describe the settlement summation of each layer. Layer settlement is the difference between initial layer thickness and the new calculated thickness. There are two components of layer settlement: resulting from layer thawing (if the layer is thawed) and from settlement as a result of thawed layer compression due to pressure on the layer.
Formula (7.20a) can be written as follows:
, where
We specify layer settlement as . It is logical to assume that . Thus . The ratio of simulated layer thickness to the initial thickness is:
The given ratio is true if the layer is completely thawed (i.e. initially completely frozen layer becomes completely melted, ).
If the layer is not subjected to thawing, it is necessary to use the following equation:
i.e. .
A more accurate result can be obtained with linear interpolation of the ratio of simulated layer thickness to the initial thickness (for partially thawed layers) as follows:
where is the initial ice content of layer ;
– stands for the unknown ice content of the layer and .
Defining layer settlement, when :
Thus:
Fig. 2 shows initial model of the ground and design points with initial parameters.
Fig. 2: Initial position of design points for settlement calculation
In Figure 2, we observe the initial temperatures, denoted , n=4. The temperatures can influence the density of the ground and therefore the settlement SNIP 2.02.0488, however, clearly does not take into account the temperature of the ground (only ice content).
CALCULATION OF SETTLEMENT CAUSED BY THE ADDITIONAL PRESSURE GENERATED BY WEIGHT OF THE STRUCTURES

(7.21) 
р_{0} is additional vertical stress on the base under the bottom of the foundation, in kPa;
b is the width of foundation base, in meters;
k_{h} is a dimensionless factor determined by the ratio z/b, where z is the distance between the foundation bottom and lower boundary of thawing zone or unsettled (during thawing) ground, in meters;
m_{th}_{,i} is compressibility factor of the ith ground layer, in kPa^{1};
k_{μ,i} is the factor determined by the ratio z_{i}_{*}/b, where z_{i}_{*} is the distance between the bottom of the foundation and the middle of the ith layer, in meters;
k_{i} and k_{i1} are the factors determined by the ratios а/b, z_{i}/b and z_{i1}/b, where z_{i} and z_{i1} are the distance from the foundation bottom to the bottom and top of the ith ground layer respectively, in meters.
Calculation Model of settlement caused by additional pressure of structures on the ground:
Fig. 3: The model of ground and structures for settlement calculation
This model allows calculating the ground settlement with the additional pressure generated by weight of structures under conditions of ground thawing.
Currently, under the described technique, we are creating a software utility that can be integrated with the Frost 3D Universal software package for calculating ground settlement. Using the geological structure of the ground created in the software and the calculated temperature field, we can determine the settlement of ground base under structures (building foundation, pipelines supports, power lines, underground pipelines, etc.) as a result of ground thawing.