# 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 stress-strain state of the base, which is described by differential equations of equilibrium and the laws of elastic-plastic 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 stress-strain state model. The main disadvantage of using numerical methods to solve a stress-strain problem is the need for big computational resources. Therefore, modern PCs are not capable of solving stress-strain 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 2.02.04-88

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 2.02.04-88, 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 non-uniform 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 2.02.04-88 "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 t0 and t respectively, and describe height of the surface as a function depending on the time h(τ), then the settlement s(t0, t)=h(t) - h(t0).

Initial time t0 represents the time, when the initial model of the problem was created (initial conditions was measured). In fact t0 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 t0, 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 2.02.04-88 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 three-dimensional, 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 L1,L2,...,Ln, 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 L1,L2,...,Ln (n=4 in our example), at the heights h1,h2,…,hn and the height of the fixed base z0 (instead of z0 the height of a design point can be used). Please note that the Li – a model of the i-th layer – may include a set of parameters (detailed below).

The model is one-dimensional and does not include ground settlement due to horizontal deformation.

## MODEL OF THE LAYER

Each homogeneous layer can be defined with a material parameter Mi a thickness hi, a temperature Ti, ice content Ii and pressures Pi at the top and Pi 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 Mi, is assumed constant. The temperature Ti, ice content Ii and height (layer thickness) hi parameters are closely related: ice content depends on the temperature, while thickness depends on ice content. In practice, the parameters Ti, Ii and hi change over the time interval. The parameter Pi does not depend on the i-th layer. Pi=Pipressure caused by the weight of i-th layer. The mass of the i-th layer only varies with water migration, if this phenomenon is not taken into account, the mass of the i-th layer is constant.

The layer model can thus be defined:

Li=(Mi,Ti,Ii,hi,Pi,Pi)

This model describes the settlement at a particular time. The initial layer model (at the initial time) can be described as follows:

 Li0=(Mi0,Ti0,Ii0,hi0,Pi0,Pi0) (7.19)

## GENERAL FORMULA FOR SETTLEMENT CALCULATION

According to SNIP 2.02.04-88, the equation for settlement is:

 s = sth + sp

where sth is the base settlement component due to the action of thawing ground weight; and

sp 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;

Ath,i and mth,i are the thawing coefficient (dimensionless) and the compressibility factor (kPa-1) of i-th layer of thawing ground (experimental data), respectively.

σzg,i is the vertical stress of ground weight at the middle of the i-th layer (kPa), determined by allowing for the buoyancy property of water. The middle of i-th layer is on the depth zi from the leveling marks;

hi is the i-th 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 i-th layer Ath,i and mth,i are simplified to: Ai:=Ath,i and mi:=mth,i, Mi:=(Ai,mi).

The initial thickness of i-th layer hi is represented with hi0, to highlight that the measurements were done at the initial time.

Instead of σzg,i (stress in the middle of the i-th layer), we will use the pressure at the top of the i-th layer Pi, and pressure below i-th layer Pi: Rewriting the formula for sth 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 2.02.04-88, 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)

р0 is additional vertical stress on the base under the bottom of the foundation, in kPa;

b is the width of foundation base, in meters;

kh 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;

mth,i is compressibility factor of the i-th ground layer, in kPa-1;

kμ,i is the factor determined by the ratio zi*/b, where zi* is the distance between the bottom of the foundation and the middle of the i-th layer, in meters;

ki and ki-1 are the factors determined by the ratios а/b, zi/b and zi-1/b, where zi and zi-1 are the distance from the foundation bottom to the bottom and top of the i-th 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.