JSACE 1 / 14 Evaluation of Base Ground Stiffness on Statically Indeterminate Framed Building Structures

Journal of Sustainable Architecture and Civil Engineering Vol. 1/ No. 14 / 2016 pp. 60-72 DOI 10.5755/j01.sace.14.1.15821 © Kaunas University of Technology Received 2016/03/24 Accepted after revision 2016/05/31 Evaluation of Base Ground Stiffness on Statically Indeterminate Framed Building Structures JSACE 1/14 Evaluation of Base Ground Stiffness on Statically Indeterminate Framed Building Structures

settlements of adjacent foundations, because when the foundations settle irregularly, additional dangerous internal forces in the structures may be caused.
On purpose to predict and avoid the negative effects of settlements of foundations, in the design process it is very important to describe and evaluate the interaction of soil particularly accurately.The mechanical behaviour of soil is very complex because of its nonlinear, heterogeneous and stress dependent nature.Therefore, in the modelling of soil, frequently particular assumptions are admitted, simplifying the complex mechanical behaviour of soil.One of the most important assumptions is Winkler's (1867) suggested model.Winkler's (1867) local elastic deformations theory describes the soil as a spring, which has a particular stiffness.That stiffness is named the modulus of subgrade reaction k s and is one of the most significant magnitudes in the calculation of the settlements of foundations.It is a conceptual magnitude, which describes the ratio between applied load and the subsequent deformation of soil (Bowles 1997).
The main issue of Winkler's model is the calculation of the modulus of subgrade reaction k s .One of the first experiments which was made on purpose to obtain this magnitude was performed by Terzaghi (1955), by loading the subgrade with a metal plate.However, these experimental results are not sufficiently accurate because of the fact that modulus of subgrade reaction depends not only on properties of soil, but also on the magnitude of load and geometrical parameters of the foundation, whereas these conditions are hardly achievable while performing the experiment with metal plates.As Ziaie-Moayed and Janbaz (2009) observed, the method of Terzaghi becomes inaccurate when dimensions of foundations are much greater than dimensions of the plate.
The modulus of subgrade reaction depends on various factors: the width of foundation's base (B), shape, the thickness of base, depth (D), the elastic modulus of material of foundation's base (E f ) and the moment of inertia (I f ), Poisson's ratio of soil (ν), the load which press the soil, the soil deformation modulus (E 0 ) and etc.The shear deformation (Poisson's) coefficient of soil describes the ratio between shear and normal stresses.The tentative magnitudes of this coefficient may be chosen by Rowe's (2001) represented proposals: for clay in undrained conditions -0,5; for clay in drained conditions -0,2-0,3; for thick sand -0,3-0,4; for powdery sand -0,1-0,3.
The modulus of subgrade reaction is a useful magnitude in modelling of foundations and subgrade interaction.As mentioned above, it does not depend on "internal" factors as the natural properties of soil, but also on "external" factors such as the geometrical characteristics of foundation or the magnitude of loading.For this reason, the modulus of subgrade reaction is not a fundamental property of soil.Depending on the mentioned external factors, for the same soil, the coefficient k s may obtain different magnitudes.There does not exist a united calculation method of the modulus of subgrade reaction.Therefore, the comparable analysis of different calculation methods was performed.Further, ten different methods of calculation of coefficient k s are analysed and evaluated according to its accuracy.
Moreover, an influence of different settlements of foundations for structures is assessed.A comparable analysis of redistribution of the internal forces of the modelled constructional elements of the structure is performed by a computer-based program.The analysis consists of comparing the redistribution of internal forces when the settlements of all foundations are the same and the case where the settlements are different.

Calculation methods
The first analysed method is the Winkler´s model.This model describes the soil as an elastic system in which there is a linear dependence between the applied load and the subsequent settlements of the foundation.The mathematical value which describes this dependence, is the modulus subgrade reaction k s : where: F-applied load to the soil; s -foundation settlement. (1) In order to determine the modulus subgrade reaction k s , the value of the settlement of the foundation (method No. 1) must be previously obtained.There are different ways to calculate these settlements.In this case, the method of addition was used, where the total settlement of the soil consists of the deformation of each small layer in which the soil is divided.
Using the principles of the elastic theory F. Schleicher (1926) presented a mathematical expression to calculate the modulus subgrade reaction (method No. 2): Vesic (1961) considered the influence of the materials and the section of the foundation and proposed an expression to obtain k s (method No. 3): where: ω -form coefficient of the foundation (for squared elastic foundation slabs -ω=0,95; for squared rigid foundation slabs -ω=0,88); E 0 -deformation modulus of soil.
where: E f -elastic modulus of foundation; I f -moment of inertia of the section of the foundation. (2) Using the principles of the elastic theory F. Schleicher (1926) presented a mathematical expression to calculate the modulus subgrade reaction (method No. 2): (2) where: ω -form coefficient of the foundation (for squared elastic foundation slabs -ω=0,95; for squared rigid foundation slabs -ω=0,88); E0 -deformation modulus of soil.Vesic (1961) considered the influence of the materials and the section of the foundation and proposed an expression to obtain ks (method No. 3): (3) The value ks' can be obtained using the next equation where: -elastic modulus of foundation; -moment of inertia of the section of the foundation.
For practical purpose Vesic reduces the expression to (method No. 4): (5) According to Timoshenko and Goodyear (1951) method to calculate foundation settlements, J. Bowles (1997)  In all of the presented methods modulus subgrade reaction ks is obtained from soil deformation modulus E0.However, this property sometimes is not known.Therefore, it can be useful to count on other kind of approximations not depending on E0.J. Bowles (1997) proposed a method in which modulus subgrade reaction ks can be obtained from the allowable bear capacity of the soil qa (method No. 6): (7) where: SF -safe factor; qa -allowable bear capacity.
Selvadurai (1984) proposed another method to obtain ks (method No. 7): The value k s ' can be obtained using the next equation Using the principles of the elastic theory F. Schleicher (1926) presented a mathematical expression to calculate the modulus subgrade reaction (method No. 2): (2) where: ω -form coefficient of the foundation (for squared elastic foundation slabs -ω=0,95; for squared rigid foundation slabs -ω=0,88); E0 -deformation modulus of soil.Vesic (1961) considered the influence of the materials and the section of the foundation and proposed an expression to obtain ks (method No. 3): (3) The value ks' can be obtained using the next equation where: -elastic modulus of foundation; -moment of inertia of the section of the foundation.
For practical purpose Vesic reduces the expression to (method No. 4): (5) According to Timoshenko and Goodyear (1951) method to calculate foundation settlements, J. Bowles (1997)  In all of the presented methods modulus subgrade reaction ks is obtained from soil deformation modulus E0.However, this property sometimes is not known.Therefore, it can be useful to count on other kind of approximations not depending on E0.J. Bowles (1997) proposed a method in which modulus subgrade reaction ks can be obtained from the allowable bear capacity of the soil qa (method No. 6): (7) where: SF -safe factor; qa -allowable bear capacity.

For practical purpose Vesic reduces the expression
Using the principles of the elastic theory F. Schleicher (1926) presented a mathematical expression to calculate the modulus subgrade reaction (method No. 2): (2) where: ω -form coefficient of the foundation (for squared elastic foundation slabs -ω=0,95; for squared rigid foundation slabs -ω=0,88); E0 -deformation modulus of soil.Vesic (1961) considered the influence of the materials and the section of the foundation and proposed an expression to obtain ks (method No. 3): (3) The value ks' can be obtained using the next equation where: -elastic modulus of foundation; -moment of inertia of the section of the foundation.
For practical purpose Vesic reduces the expression to (method No. 4): (5) According to Timoshenko and Goodyear (1951) method to calculate foundation settlements, J. Bowles (1997)  In all of the presented methods modulus subgrade reaction ks is obtained from soil deformation modulus E0.However, this property sometimes is not known.Therefore, it can be useful to count on other kind of approximations not depending on E0.J. Bowles (1997) proposed a method in which modulus subgrade reaction ks can be obtained from the allowable bear capacity of the soil qa (method No. 6): (7) where: SF -safe factor; qa -allowable bear capacity.Selvadurai (1984) proposed another method to obtain ks (method No. 7): Using the principles of the elastic theory F. Schleicher (1926) presented a mathematical expression to calculate the modulus subgrade reaction (method No. 2): (2) where: ω -form coefficient of the foundation (for squared elastic foundation slabs -ω=0,95; for squared rigid foundation slabs -ω=0,88); E0 -deformation modulus of soil.Vesic (1961) considered the influence of the materials and the section of the foundation and proposed an expression to obtain ks (method No. 3): (3) The value ks' can be obtained using the next equation where: -elastic modulus of foundation; -moment of inertia of the section of the foundation.
For practical purpose Vesic reduces the expression to (method No. 4): (5) According to Timoshenko and Goodyear (1951) method to calculate foundation settlements, J. Bowles (1997) proposed another equation to obtain ks (method No. 5): (6) where: IF -influence factor, which depends on the ratio D/L and L/B (L-foundation length; Bfoundation width; D -foundation slab depth) and on Poisson's ratio ν; IS -influence factor, which depends on L'/B' ratio (L' and B' -effective foundation dimensions), width of the soil layer, Poisson's coefficient ν and depth of the foundation D. m -number of corners contributing to settlement (at a corner of the footing m=1; at the centre m=4).
In all of the presented methods modulus subgrade reaction ks is obtained from soil deformation modulus E0.However, this property sometimes is not known.Therefore, it can be useful to count on other kind of approximations not depending on E0.J. Bowles (1997) proposed a method in which modulus subgrade reaction ks can be obtained from the allowable bear capacity of the soil qa (method No. 6): (7) where: SF -safe factor; qa -allowable bear capacity.

 
Using the principles of the elastic theory F. Schleicher (1926) presented a mathematical expression to calculate the modulus subgrade reaction (method No. 2): (2) where: ω -form coefficient of the foundation (for squared elastic foundation slabs -ω=0,95; for squared rigid foundation slabs -ω=0,88); E0 -deformation modulus of soil.Vesic (1961) considered the influence of the materials and the section of the foundation and proposed an expression to obtain ks (method No. 3): (3) The value ks' can be obtained using the next equation where: -elastic modulus of foundation; -moment of inertia of the section of the foundation.
For practical purpose Vesic reduces the expression to (method No. 4): (5) According to Timoshenko and Goodyear (1951) method to calculate foundation settlements, J. Bowles (1997)  In all of the presented methods modulus subgrade reaction ks is obtained from soil deformation modulus E0.However, this property sometimes is not known.Therefore, it can be useful to count on other kind of approximations not depending on E0.J. Bowles (1997) proposed a method in which modulus subgrade reaction ks can be obtained from the allowable bear capacity of the soil qa (method No. 6): (7) where: SF -safe factor; qa -allowable bear capacity.
Selvadurai (1984) proposed another method to obtain ks (method No. 7): According to Timoshenko and Goodyear (1951) method to calculate foundation settlements, J. Bowles (1997) proposed another equation to obtain k s (method No. 5): Using the principles of the elastic theory F. Schleicher (1926) presented a mathematical expression to calculate the modulus subgrade reaction (method No. 2): (2) where: ω -form coefficient of the foundation (for squared elastic foundation slabs -ω=0,95; for squared rigid foundation slabs -ω=0,88); E0 -deformation modulus of soil.Vesic (1961) considered the influence of the materials and the section of the foundation and proposed an expression to obtain ks (method No. 3): (3) The value ks' can be obtained using the next equation where: -elastic modulus of foundation; -moment of inertia of the section of the foundation.
For practical purpose Vesic reduces the expression to (method No. 4): (5) According to Timoshenko and Goodyear (1951) method to calculate foundation settlements, J. Bowles (1997)  In all of the presented methods modulus subgrade reaction ks is obtained from soil deformation modulus E0.However, this property sometimes is not known.Therefore, it can be useful to count on other kind of approximations not depending on E0.J. Bowles (1997) proposed a method in which modulus subgrade reaction ks can be obtained from the allowable bear capacity of the soil qa (method No. 6): (7) where: SF -safe factor; qa -allowable bear capacity.where: SF -safe factor; q a -allowable bear capacity.

 
In all of the presented methods modulus subgrade reaction k s is obtained from soil deformation modulus E 0 .However, this property sometimes is not known.Therefore, it can be useful to count on other kind of approximations not depending on E 0 .J. Bowles (1997) proposed a method in which modulus subgrade reaction k s can be obtained from the allowable bear capacity of the soil q a (method No. 6): (7) Selvadurai (1984) proposed another method to obtain k s (method No. 7): (6) where: IF -influence factor, which depends on the ratio D/L and L/B (L-foundation length; Bfoundation width; D -foundation slab depth) and on Poisson's ratio ν; IS -influence factor, which depends on L'/B' ratio (L' and B' -effective foundation dimensions), width of the soil layer, Poisson's coefficient ν and depth of the foundation D. m -number of corners contributing to settlement (at a corner of the footing m=1; at the centre m=4).
In all of the presented methods modulus subgrade reaction ks is obtained from soil deformation modulus E0.However, this property sometimes is not known.Therefore, it can be useful to count on other kind of approximations not depending on E0.J. Bowles (1997) proposed a method in which modulus subgrade reaction ks can be obtained from the allowable bear capacity of the soil qa (method No. 6): (7) where: SF -safe factor; qa -allowable bear capacity.Selvadurai (1984) proposed another method to obtain ks (method No. 7): (8) Modulus ks can be obtained using finite-element analysis software Autodesk Robot Structural Analysis Professional (further -Autodesk Robot) (method No. 8).This program obtains ks by predicting the settlements of a footing by the method of addition (which is used in the method No. 1).

 
Modulus k s can be obtained using finite-element analysis software Autodesk Robot Structural Analysis Professional (further -Autodesk Robot) (method No. 8).This program obtains k s by predicting the settlements of a footing by the method of addition (which is used in the method No. 1).
Another way to determine modulus of subgrade reaction is to apply the finite layer method.It is assumed that, under a footing, there are vertical shear forces between the particles of the soil, so that the soil behaves plastically.However, going deeper in the ground, it occurs that these shear forces become weaker and, at some depth, disappear.Then, the considered soil behaves elastically.This means that there is a layer under the footing of thickness H sl in which there are plastic deformations.It is recommended to take H sl =1/4 B. The results obtained by some authors (Gorbunov-Posadov et al. (1984), Zhemochkin B. N. andSinitsyn A. P. (1947)) are provided in ( 8) and ( 9) equations: In all of the presented methods modulus subgrade reaction ks is obtained from soil deformation modulus E0.However, this property sometimes is not known.Therefore, it can be useful to count on other kind of approximations not depending on E0.J. Bowles (1997) proposed a method in which modulus subgrade reaction ks can be obtained from the allowable bear capacity of the soil qa (method No. 6): (7) where: SF -safe factor; qa -allowable bear capacity.Selvadurai (1984) proposed another method to obtain ks (method No. 7): (8) Modulus ks can be obtained using finite-element analysis software Autodesk Robot Structural Analysis Professional (further -Autodesk Robot) (method No. 8).This program obtains ks by predicting the settlements of a footing by the method of addition (which is used in the method No. 1).

 
The first equation (Eq.9) considers that the plastic layer can slide on the ground, which is under it (method No. 9); the other equation assumes that slip does not occur (method No. 10).
Another way to determine modulus of subgrade reaction is to apply the finite layer method.It is assumed that, under a footing, there are vertical shear forces between the particles of the soil, so that the soil behaves plastically.However, going deeper in the ground, it occurs that these shear forces become weaker and, at some depth, disappear.Then, the considered soil behaves elastically.This means that there is a layer under the footing of thickness Hsl in which there are plastic deformations.It is recommended to take Hsl=1/4 B. The results obtained by some authors (Gorbunov-Posadov et al. (1984), Zhemochkin B. N. and Sinitsyn A. P. ( 1947)) are provided in (8) and (9) equations: The first equation (Eq.9) considers that the plastic layer can slide on the ground, which is under it (method No. 9); the other equation assumes that slip does not occur (method No. 10).

Calculation of modulus subgrade reaction for a multi-layer soil
In most of analysed methods, modulus ks depends on other ground properties like deformation modulus E0, Poisson's coefficient ν and bear capacity qa.When under the footing is a one-layered soil, these properties are assumed be uniform and the modulus ks can be obtained simply using the considered methods.
Nevertheless, frequently ground consists of several layers with different type of soil, mechanical properties, granulometric composition, thickness, water saturation degree and etc.Therefore, in order to calculate modulus subgrade reaction using the analysed methods and considering these different properties between soil layers, authors propose to apply the influence factor ki to each soil layer.This coefficient depends on the thickness and the depth of the considered layer.The thinner and deeper layer is the less significant layer.The influence factor ki also depends on the deformation modulus E0 of the layer and on the form and dimensions of the foundation.
First of all transitional coefficient ktr,i is obtained: Another way to determine modulus of subgrade reaction is to apply the finite layer method.It is assumed that, under a footing, there are vertical shear forces between the particles of the soil, so that the soil behaves plastically.However, going deeper in the ground, it occurs that these shear forces become weaker and, at some depth, disappear.Then, the considered soil behaves elastically.This means that there is a layer under the footing of thickness Hsl in which there are plastic deformations.It is recommended to take Hsl=1/4 B. The results obtained by some authors (Gorbunov-Posadov et al. (1984), Zhemochkin B. N. and Sinitsyn A. P. ( 1947)) are provided in ( 8) and (9) equations: The first equation (Eq.9) considers that the plastic layer can slide on the ground, which is under it (method No. 9); the other equation assumes that slip does not occur (method No. 10).

Calculation of modulus subgrade reaction for a multi-layer soil
In most of analysed methods, modulus ks depends on other ground properties like deformation modulus E0, Poisson's coefficient ν and bear capacity qa.When under the footing is a one-layered soil, these properties are assumed be uniform and the modulus ks can be obtained simply using the considered methods.
Nevertheless, frequently ground consists of several layers with different type of soil, mechanical properties, granulometric composition, thickness, water saturation degree and etc.Therefore, in order to calculate modulus subgrade reaction using the analysed methods and considering these different properties between soil layers, authors propose to apply the influence factor ki to each soil layer.This coefficient depends on the thickness and the depth of the considered layer.The thinner and deeper layer is the less significant layer.The influence factor ki also depends on the deformation modulus E0 of the layer and on the form and dimensions of the foundation.
First of all transitional coefficient ktr,i is obtained: In most of analysed methods, modulus k s depends on other ground properties like deformation modulus E 0 , Poisson's coefficient ν and bear capacity q a .When under the footing is a one-layered soil, these properties are assumed be uniform and the modulus k s can be obtained simply using the considered methods.
Nevertheless, frequently ground consists of several layers with different type of soil, mechanical properties, granulometric composition, thickness, water saturation degree and etc.Therefore, in order to calculate modulus subgrade reaction using the analysed methods and considering these different properties between soil layers, authors propose to apply the influence factor k i to each soil layer.This coefficient depends on the thickness and the depth of the considered layer.The Calculation of modulus subgrade reaction for a multi-layer soil thinner and deeper layer is the less significant layer.The influence factor k i also depends on the deformation modulus E 0 of the layer and on the form and dimensions of the foundation.
First of all transitional coefficient k tr,i is obtained: The transitional coefficient may be obtained from the tables presented by some authors (e. g., Šližytė et al. (2012)).In this case k tr,i can be selected from the ratio between foundation (10) The first equation (Eq.9) considers that the plastic layer can slide on the ground, which is under it (method No. 9); the other equation assumes that slip does not occur (method No. 10).

Calculation of modulus subgrade reaction for a multi-layer soil
In most of analysed methods, modulus ks depends on other ground properties like deformation modulus E0, Poisson's coefficient ν and bear capacity qa.When under the footing is a one-layered soil, these properties are assumed be uniform and the modulus ks can be obtained simply using the considered methods.
Nevertheless, frequently ground consists of several layers with different type of soil, mechanical properties, granulometric composition, thickness, water saturation degree and etc.Therefore, in order to calculate modulus subgrade reaction using the analysed methods and considering these different properties between soil layers, authors propose to apply the influence factor ki to each soil layer.This coefficient depends on the thickness and the depth of the considered layer.The thinner and deeper layer is the less significant layer.The influence factor ki also depends on the deformation modulus E0 of the layer and on the form and dimensions of the foundation.
First of all transitional coefficient ktr,i is obtained: (11) where: L1 -half of the length of the foundation; B1 -half of the foundation width; zi -upper depth of the considered layer.
(12) The transitional coefficient may be obtained from the tables presented by some authors (e. g., Šližytė et al. (2012)).In this case ktr,i can be selected from the ratio between foundation and from the relative depth of the analysed layer .Transitional coefficients ktr,i are normalized and the influence factor of layer is obtained: The final joint properties of the overall multi-layered soil are calculated by these equations: and from the relative depth of the analysed layer (10) The first equation (Eq.9) considers that the plastic layer can slide on the ground, which is under it (method No. 9); the other equation assumes that slip does not occur (method No. 10).

Calculation of modulus subgrade reaction for a multi-layer soil
In most of analysed methods, modulus ks depends on other ground properties like deformation modulus E0, Poisson's coefficient ν and bear capacity qa.When under the footing is a one-layered soil, these properties are assumed be uniform and the modulus ks can be obtained simply using the considered methods.
Nevertheless, frequently ground consists of several layers with different type of soil, mechanical properties, granulometric composition, thickness, water saturation degree and etc.Therefore, in order to calculate modulus subgrade reaction using the analysed methods and considering these different properties between soil layers, authors propose to apply the influence factor ki to each soil layer.This coefficient depends on the thickness and the depth of the considered layer.The thinner and deeper layer is the less significant layer.The influence factor ki also depends on the deformation modulus E0 of the layer and on the form and dimensions of the foundation.
First of all transitional coefficient ktr,i is obtained: (11) where: L1 -half of the length of the foundation; B1 -half of the foundation width; zi -upper depth of the considered layer.
(12) The transitional coefficient may be obtained from the tables presented by some authors (e. g., Šližytė et al. (2012)).In this case ktr,i can be selected from the ratio between foundation and from the relative depth of the analysed layer .Transitional coefficients ktr,i are normalized and the influence factor of layer is obtained: The final joint properties of the overall multi-layered soil are calculated by these equations: . Transitional coefficients k tr,i are normalized and the influence factor of layer is obtained: modulus E0, Poisson's coefficient ν and bear capacity qa.When under the footing is a one-layered soil, these properties are assumed be uniform and the modulus ks can be obtained simply using the considered methods.
Nevertheless, frequently ground consists of several layers with different type of soil, mechanical properties, granulometric composition, thickness, water saturation degree and etc.Therefore, in order to calculate modulus subgrade reaction using the analysed methods and considering these different properties between soil layers, authors propose to apply the influence factor ki to each soil layer.This coefficient depends on the thickness and the depth of the considered layer.The thinner and deeper layer is the less significant layer.The influence factor ki also depends on the deformation modulus E0 of the layer and on the form and dimensions of the foundation.
First of all transitional coefficient ktr,i is obtained: (11) where: L1 -half of the length of the foundation; B1 -half of the foundation width; zi -upper depth of the considered layer.
(12) The transitional coefficient may be obtained from the tables presented by some authors (e. g., Šližytė et al. (2012)).In this case ktr,i can be selected from the ratio between foundation and from the relative depth of the analysed layer .Transitional coefficients ktr,i are normalized and the influence factor of layer is obtained: The final joint properties of the overall multi-layered soil are calculated by these equations: The first equation (Eq.9) considers that the plastic layer can slide on the ground, which is under it (method No. 9); the other equation assumes that slip does not occur (method No. 10).

Calculation of modulus subgrade reaction for a multi-layer soil
In most of analysed methods, modulus ks depends on other ground properties like deformation modulus E0, Poisson's coefficient ν and bear capacity qa.When under the footing is a one-layered soil, these properties are assumed be uniform and the modulus ks can be obtained simply using the considered methods.
Nevertheless, frequently ground consists of several layers with different type of soil, mechanical properties, granulometric composition, thickness, water saturation degree and etc.Therefore, in order to calculate modulus subgrade reaction using the analysed methods and considering these different properties between soil layers, authors propose to apply the influence factor ki to each soil layer.This coefficient depends on the thickness and the depth of the considered layer.The thinner and deeper layer is the less significant layer.The influence factor ki also depends on the deformation modulus E0 of the layer and on the form and dimensions of the foundation.
First of all transitional coefficient ktr,i is obtained: (11) where: L1 -half of the length of the foundation; B1 -half of the foundation width; zi -upper depth of the considered layer.
(12) The transitional coefficient may be obtained from the tables presented by some authors (e. g., Šližytė et al. (2012)).In this case ktr,i can be selected from the ratio between foundation and from the relative depth of the analysed layer .Transitional coefficients ktr,i are normalized and the influence factor of layer is obtained: The final joint properties of the overall multi-layered soil are calculated by these equations: The final joint properties of the overall multi-layered soil are calculated by these equations: (10) The first equation (Eq.9) considers that the plastic layer can slide on the ground, which is under it (method No. 9); the other equation assumes that slip does not occur (method No. 10).

Calculation of modulus subgrade reaction for a multi-layer soil
In most of analysed methods, modulus ks depends on other ground properties like deformation modulus E0, Poisson's coefficient ν and bear capacity qa.When under the footing is a one-layered soil, these properties are assumed be uniform and the modulus ks can be obtained simply using the considered methods.
Nevertheless, frequently ground consists of several layers with different type of soil, mechanical properties, granulometric composition, thickness, water saturation degree and etc.Therefore, in order to calculate modulus subgrade reaction using the analysed methods and considering these different properties between soil layers, authors propose to apply the influence factor ki to each soil layer.This coefficient depends on the thickness and the depth of the considered layer.The thinner and deeper layer is the less significant layer.The influence factor ki also depends on the deformation modulus E0 of the layer and on the form and dimensions of the foundation.
First of all transitional coefficient ktr,i is obtained: (11) where: L1 -half of the length of the foundation; B1 -half of the foundation width; zi -upper depth of the considered layer.
(12) The transitional coefficient may be obtained from the tables presented by some authors (e. g., Šližytė et al. (2012)).In this case ktr,i can be selected from the ratio between foundation and from the relative depth of the analysed layer .Transitional coefficients ktr,i are normalized and the influence factor of layer is obtained: The final joint properties of the overall multi-layered soil are calculated by these equations: The first equation (Eq.9) considers that the plastic layer can slide on the ground, which is under it (method No. 9); the other equation assumes that slip does not occur (method No. 10).

Calculation of modulus subgrade reaction for a multi-layer soil
In most of analysed methods, modulus ks depends on other ground properties like deformation modulus E0, Poisson's coefficient ν and bear capacity qa.When under the footing is a one-layered soil, these properties are assumed be uniform and the modulus ks can be obtained simply using the considered methods.
Nevertheless, frequently ground consists of several layers with different type of soil, mechanical properties, granulometric composition, thickness, water saturation degree and etc.Therefore, in order to calculate modulus subgrade reaction using the analysed methods and considering these different properties between soil layers, authors propose to apply the influence factor ki to each soil layer.This coefficient depends on the thickness and the depth of the considered layer.The thinner and deeper layer is the less significant layer.The influence factor ki also depends on the deformation modulus E0 of the layer and on the form and dimensions of the foundation.
First of all transitional coefficient ktr,i is obtained: (11) where: L1 -half of the length of the foundation; B1 -half of the foundation width; zi -upper depth of the considered layer.
(12) The transitional coefficient may be obtained from the tables presented by some authors (e. g., Šližytė et al. (2012)).In this case ktr,i can be selected from the ratio between foundation and from the relative depth of the analysed layer .Transitional coefficients ktr,i are normalized and the influence factor of layer is obtained: The final joint properties of the overall multi-layered soil are calculated by these equations: Applying this method, the significant ground depth is . For weak soils it can be adopted .

Analysis of the different methods
In order to check the accuracy of the presented methods, the modulus subgrade reaction was obtained from four different geologies boreholes.Two of them (geologies A and B) are one -layered soils and the other two (geologies C and D) are multi-layered soils.For each one of them, the modulus of subgrade reaction ks is calculated, applying three different vertical loads to a square shallow footing.The loads are: 500 kN, 1000 kN and 2000 kN.The dimensions of the square shallow foundation are selected in each case according to soil parameters and the magnitude of load, in the Applying this method, the significant ground depth is . For weak soils it can be adopted .

Analysis of the different methods
In order to check the accuracy of the presented methods, the modulus subgrade reaction was obtained from four different geologies boreholes.Two of them (geologies A and B) are one -layered soils and the other two (geologies C and D) are multi-layered soils.For each one of them, the modulus of subgrade reaction ks is calculated, applying three different vertical loads to a square shallow footing.The loads are: 500 kN, 1000 kN and 2000 kN.The dimensions of the square shallow Applying this method, the significant ground depth is B , H ⋅ = 5 1 . For weak soils it can be adopted In order to check the accuracy of the presented methods, the modulus subgrade reaction was obtained from four different geologies boreholes.Two of them (geologies A and B) are one -layered soils and the other two (geologies C and D) are multi-layered soils.For each one of them, the modulus of subgrade reaction k s is calculated, applying three different vertical loads to a square shallow footing.The loads are: 500 kN, 1000 kN and 2000 kN.The dimensions of the square shallow foundation are selected in each case according to soil parameters and the magnitude of load, in the way that bearing capacity would not be exceeded (dimensions of foundations vary from 1,3 m to 3,6 m).
The properties of each geological layer are presented in table 1.The calculation of modulus k s was carried out for all 4 geological situations.The results were obtained using the 10 different analysed methods, which are shown in Fig. 1 and 2. Winkler's method (method No. 1) is assumed to be the standard with which the other method can be compared.

Analysis of the different methods
The results obtained by methods No. 9 and 10 differed considerably: k s values were about 4 times where: H sl -layer thickness; q c -cone tip resistance; E 0 -deformation modulus; c' -effective cohesion; φsoil friction angle; γ -weight; ν -Poisson's coefficient.
bigger than those obtained using the other methods.As a consequence, methods No. 9 and 10 were not analysed.In figure 1 ks values of one-layered soils A (sandy soil) and B (clayey soil) are shown, calculated using all the discussed methods.In both cases the calculations show common accuracy trends.The obtained results are quite similar, however using the reduction of Vesic's equation (method No. 3) and Salvadurai's expression (method No. 7), ks values are the lowest.It means that by these methods, coefficient ks is calculated too carefully.In the case of the second method of Bowles (method No. 6) the difference of the obtained ks values, applying different magnitude loads, is the highest.

Fig. 2. Values for soil C (left) and soil D (right)
In figure 2 it is presented the subgrade modulus values (using 8 different methods) for soils C and D. Soil C and D are composed of several different layers.Therefore, influence factor ki is included in the calculations of modulus ks.In one-layered soil diagram (figure 1) it can be seen the resembling trends: using methods No. 3 and 7, the results keep a reserve (are not considerable); the results of method No. 6 are strongly dependent of the magnitude of the applied load.
Using the method No. 1, footing settlements were predicted by the addition method, and then the modulus of subgrade reaction ks was calculated (eq.1).Modulus ks is obtained directly from the definition of Winkler's theory.On the other hand, the other methods are approximations from which In Fig. 1 k s values of one-layered soils A (sandy soil) and B (clayey soil) are shown, calculated using all the discussed methods.In both cases the calculations show common accuracy trends.The obtained results are quite similar, however using the reduction of Vesic's equation (method No. 3) and Salvadurai's expression (method No. 7), k s values are the lowest.It means that by these methods, coefficient k s is calculated too carefully.In the case of the second method of Bowles (method No. 6) the difference of the obtained k s values, applying different magnitude loads, is the highest.In figure 1 ks values of one-layered soils A (sandy soil) and B (clayey soil) are shown, calculated using all the discussed methods.In both cases the calculations show common accuracy trends.The obtained results are quite similar, however using the reduction of Vesic's equation (method No. 3) and Salvadurai's expression (method No. 7), ks values are the lowest.It means that by these methods, coefficient ks is calculated too carefully.In the case of the second method of Bowles (method No. 6) the difference of the obtained ks values, applying different magnitude loads, is the highest.

Fig. 2. Values for soil C (left) and soil D (right)
In figure 2 it is presented the subgrade modulus values (using 8 different methods) for soils C and D. Soil C and D are composed of several different layers.Therefore, influence factor ki is included in the calculations of modulus ks.In one-layered soil diagram (figure 1) it can be seen the resembling trends: using methods No. 3 and 7, the results keep a reserve (are not considerable); the results of method No. 6 are strongly dependent of the magnitude of the applied load.
Using the method No. 1, footing settlements were predicted by the addition method, and then the modulus of subgrade reaction ks was calculated (eq.1).Modulus ks is obtained directly from the definition of Winkler's theory.On the other hand, the other methods are approximations from which In Fig. 2 it is presented the subgrade modulus values (using 8 different methods) for soils C and D.
Soil C and D are composed of several different layers.Therefore, influence factor k i is included in the calculations of modulus k s .In one-layered soil diagram (Fig. 1) it can be seen the resembling trends: using methods No. 3 and 7, the results keep a reserve (are not considerable); the results of method No. 6 are strongly dependent of the magnitude of the applied load.
Using the method No. 1, footing settlements were predicted by the addition method, and then the modulus of subgrade reaction k s was calculated (eq.1).Modulus k s is obtained directly from the definition of Winkler's theory.On the other hand, the other methods are approximations from which the modulus k s is calculated using the other soil and footing properties.Therefore, method No. 1 is assumed to be the standard with which the other methods can be compared in order to determine its accuracy.
The difference between each method and Winkler's (method No. 1) is calculated from the next equation: (17) the modulus ks is calculated using the other soil and footing properties.Therefore, method No. 1 is assumed to be the standard with which the other methods can be compared in order to determine its accuracy.
The difference between each method and Winkler's (method No. 1) is calculated from the next equation: (17) Difference Δ was calculated for each one of the soils and for each one of loading cases (500 kN, 1000 kN and 2000 kN), applying the different ks calculation methods.The unified Δ value for each soil is obtained calculating average of the obtained values in each loading case.Finally average Δ value for one-layered soils (A and B) and average Δ value for multi-layered soils (C and D) were obtained.The comparison analysis of these final Δ results is presented in figure 5.In figure 3 it can be noticed that the closest results to Method 1 are obtained with methods No. 8 (Autodesk Robot) and No. 5 (J.Bowles).In the case of one-layered (A and B) soils these methods are very precise.Difference Δ reach 6 % and 8 %, respectively.In the case of multi-layered (C and D) soils, Δ is more considerable: for both is about 12 %.As far as these two mentioned methods (No. 8 and No. 5) are the most exact paths to obtain ks, it is recommended to apply them in the design of building foundations.
Also in figure 3 it can be seen that when the layer influence factor ki in the calculations of multi-layered soils is introduced, ks accuracy (Δ) values changes (in comparison with the case of onelayered soils).For some methods (No. 5 and 8) the difference Δ grows about 5-8 %.In other cases (method No. 3,4,6 and 7), the difference Δ decreases up to 10 % (in this case, the introduction of influence factor ki increases the accuracy of the results).In method No. 2, Δ remains similar: it just increases for multi-layered soils up to 2 %.Broadly speaking, using any calculation method, the values of Δ between one-layered and multi-layered soil do not differ more than 10 %.Therefore, the use of layer influence factor ki is justified.

Constructional analysis
Structural analysis was carried out, using program Autodesk Robot.The goal was to explore the changes of internal forces in the structures, using modulus of subgrade reaction obtained by different methods.There were three methods used in the analysis: Bowles's equation (using influence  the modulus ks is calculated using the other soil and footing properties.Therefore, method No. 1 is assumed to be the standard with which the other methods can be compared in order to determine its accuracy. The difference between each method and Winkler's (method No. 1) is calculated from the next equation: (17) Difference Δ was calculated for each one of the soils and for each one of loading cases (500 kN, 1000 kN and 2000 kN), applying the different ks calculation methods.The unified Δ value for each soil is obtained calculating average of the obtained values in each loading case.Finally average Δ value for one-layered soils (A and B) and average Δ value for multi-layered soils (C and D) were obtained.The comparison analysis of these final Δ results is presented in figure 5.In figure 3 it can be noticed that the closest results to Method 1 are obtained with methods No. 8 (Autodesk Robot) and No. 5 (J.Bowles).In the case of one-layered (A and B) soils these methods are very precise.Difference Δ reach 6 % and 8 %, respectively.In the case of multi-layered (C and D) soils, Δ is more considerable: for both is about 12 %.As far as these two mentioned methods (No. 8 and No. 5) are the most exact paths to obtain ks, it is recommended to apply them in the design of building foundations.
Also in figure 3 it can be seen that when the layer influence factor ki in the calculations of multi-layered soils is introduced, ks accuracy (Δ) values changes (in comparison with the case of onelayered soils).For some methods (No. 5 and 8) the difference Δ grows about 5-8 %.In other cases (method No. 3, 4, 6 and 7), the difference Δ decreases up to 10 % (in this case, the introduction of influence factor ki increases the accuracy of the results).In method No. 2, Δ remains similar: it just increases for multi-layered soils up to 2 %.Broadly speaking, using any calculation method, the values of Δ between one-layered and multi-layered soil do not differ more than 10 %.Therefore, the use of layer influence factor ki is justified.

Constructional analysis
Structural analysis was carried out, using program Autodesk Robot.The goal was to explore the changes of internal forces in the structures, using modulus of subgrade reaction obtained by different methods.There were three methods used in the analysis: Bowles's equation (using influence

 
In Fig. 3 it can be noticed that the closest results to Method 1 are obtained with methods No. 8 (Autodesk Robot) and No. 5 (J.Bowles).In the case of one-layered (A and B) soils these methods are very precise.Difference Δ reach 6 % and 8 %, respectively.In the case of multi-layered (C and D) soils, Δ is more considerable: for both is about 12 %.As far as these two mentioned methods (No. 8 and No. 5) are the most exact paths to obtain k s , it is recommended to apply them in the design of building foundations.
Also in Fig. 3 it can be seen that when the layer influence factor k i in the calculations of multi-layered soils is introduced, k s accuracy (Δ) values changes (in comparison with the case of one-layered soils).For some methods (No. 5 and 8) the difference Δ grows about 5-8 %.In other cases (method No. 3, 4, 6 and 7), the difference Δ decreases up to 10 % (in this case, the introduction of influence factor k i increases the accuracy of the results).In method No. 2, Δ remains similar: it just increases for multi-layered soils up to 2 %.Broadly speaking, using any calculation method, the values of Δ between one-layered and multi-layered soil do not differ more than 10 %.Therefore, the use of layer influence factor k i is justified.
Structural analysis was carried out, using program Autodesk Robot.The goal was to explore the changes of internal forces in the structures, using modulus of subgrade reaction obtained by different methods.There were three methods used in the analysis: Bowles's equation (using influence factor for multi-layered soil) (method No. 5), calculation with Autodesk Robot (method where: H sl -layer thickness; q c -cone tip resistance; E 0 -deformation modulus; c' -effective cohesion; φ -soil friction angle; γ -weight; ν -Poisson's coefficient.

Diagrams of bending moments
Table 3 Legend of the diagrams   In figure 5, it can be noticed that, when the springs are introduced (situation No. 2) instead of rigid supports (situation 1), bending moments in the beams increase up to 43 %.The maximum increase of bending moments can be noticed in beams B1 and B5, in the places, where those beams are connected to columns C1 and C6 respectively.However, the values of bending moments in several beams decrease.The biggest decline of bending moments is noticed in the places, where beams are connected to columns C2 and C5.In these places the values of bending moments decreased by up to 21 %.There are also significant differences of bending moments in the columns.Bending moments increased by up to 41 % at the top of the columns C1 and C6.
Introducing different types of soil (situation No. 3) and comparing it to the results obtained from situation No. 2, where there is only one type of soil, changes are also noticeable.The maximum increase of bending moments, 72 %, can be noticed in beam B1, in the place where this beam is connected to column C2.Comparing the latter situations, there is a considerable decrease of bending moments in some structures.For example, the bending moment in the midspan of beam B1 was up to 21 %.Although the changes of bending moments in columns are not so significant, they should also be evaluated.The most noticeable decrease, 23 % occurred at the top of column C1.The increase 3) and comparing it to the results obtained from situation No. 2, where there is only one type of soil, changes are also noticeable.The maximum increase of bending moments, 72 %, can be noticed in beam B1, in the place where this beam is connected to column C2.Comparing the latter situations, there is a considerable decrease of bending moments in some structures.For example, the bending moment in the midspan of beam B1 was up to 21 %.Although the changes of bending moments in columns are not so significant, they should also be evaluated.
The most noticeable decrease, 23 % occurred at the top of column C1.The increase of bending moments in columns is not considerable.
When the springs are introduced (situation No. 2) instead of rigid supports (situation No. 1), the lateral forces in the beams increase by up to 7 %.This maximum increase is noticed in beams B1 and B5, in the connections to columns C1 and C6.The decrease of lateral forces in some of the beams is also considerable.Maximum decrease is noticed in beam B2, in the connection to column C2.This maximum decrease reaches 6 %.When comparing situation No. 2 to situation No. 1, it can be seen, that lateral forces increase in all of the columns of the first floor (columns C1 to C6).The maximum increase, up to 38 %, occurs in columns C1 and C6.
Comparing the values of the lateral forces obtained from situation No. 3 to the results obtained from situation No. 3, significant changes can be noticed.The maximum increase of lateral forces occurs in beam B2, in its connection to column C2.The values obtained in situation No. 3 are 21 % larger than in situation No. 2. The maximum decrease is also in beam B2.This decrease is noted at the place where the beam B2 is connected to the column C3.The values obtained in situation No. 3 decrease 18 % in comparison to situation No. 2. While comparing the changes of lateral forces in columns under situation No. 3 to situation No. 2, the maximum increase, up to 28 %, can be noticed in columns C4 and C5.The maximum decrease also occurs in the same columns.It is possible because there are many load combinations.This leads to the fact that there are diagrams on both sides of the columns.Lateral forces increase on one side and decrease on the other side. 1 The comparable analysis of calculation methods of the modulus of subgrade reaction was performed.When analysing it was assumed that the most realistic results are obtained with the first method (Winkler's).The accuracy of other methods was determined by comparing those methods with the method No. 1 in calculating the difference Δ.The closest results to first method were derived from calculation methods No. 5 (Bowles) and No. 8 (Autodesk Robot) where the difference Δ does not exceed 12 %.Applying methods No. 2, 4 and 6 the value difference Δ oscillate vary from 12 % to 25 %.The largest difference from the first method is obtained calculating by methods No. 3 (simplified Vesic's equation) and No. 7 (Selvadurai) when the difference Δ reaches 48 %.
2 Most of the considered methods are dedicated to homogenous soils.However, in practise often occurs that the soil is consisted of many layers, which have different properties.In order to calculate the modulus of subgrade reaction of such soil, it is proposed to use the layer influence factor k i .By some methods (method No. 5 and 8) the difference of methods increased in one-layer case by 5 %.The difference Δ of other methods remained similar or even less than Δ of these methods in one-layer case.The differences Δ, while calculating the coefficient k s by the same method for one-layered and multi-layered soil, never differed more than 10 %.Therefore, the application of the coefficient k i is advisable and rational in the determination of the modulus of subgrade reaction of a multi-layered soil.
3 When the analysis of considered building was performed, it was determined that eval- uation of the modulus of subgrade reaction has considerable influence for the internal forces of the structures.Redistribution of internal forces occurred with particular intensity when there were modelled different base ground stiffness (situation No. 3) under supports S1 and S2.Redistribution of bending moments was the most relevant: under situation No. 3 it increased more than 70 %.Moreover, there was a considerable variation of shear forces in the columns.Evaluating the base ground stiffness (situation No. 2) and comparing it with the internal forces of the case where the supports are rigid (situation No. 1), the increase reached even 40 %.In beams considerable changes of shear forces were determined introducing the different types of soil (situation No. 3).It exceeded 20 %.The changes of axial forces were not such considerable (up to 10 %).However, the evaluation of those changes is essential.
4 The analysis of structures was performed evaluating the different modulus of subgrade reaction in several cases: Bowles's method (No. 5), Winkler's method (No. 1) and Autodesk Robot method (No. 8).Obtained internal forces were compared in all cases.Calculating the stiff-

Conclusions
proposed another equation to obtain ks (method No. 5): (6) where: IF -influence factor, which depends on the ratio D/L and L/B (L-foundation length; Bfoundation width; D -foundation slab depth) and on Poisson's ratio ν; IS -influence factor, which depends on L'/B' ratio (L' and B' -effective foundation dimensions), width of the soil layer, Poisson's coefficient ν and depth of the foundation D. m -number of corners contributing to settlement (at a corner of the footing m=1; at the centre m=4).
proposed another equation to obtain ks (method No. 5): (6) where: IF -influence factor, which depends on the ratio D/L and L/B (L-foundation length; Bfoundation width; D -foundation slab depth) and on Poisson's ratio ν; IS -influence factor, which depends on L'/B' ratio (L' and B' -effective foundation dimensions), width of the soil layer, Poisson's coefficient ν and depth of the foundation D. m -number of corners contributing to settlement (at a corner of the footing m=1; at the centre m=4).
proposed another equation to obtain ks (method No. 5): (6) where: IF -influence factor, which depends on the ratio D/L and L/B (L-foundation length; Bfoundation width; D -foundation slab depth) and on Poisson's ratio ν; IS -influence factor, which depends on L'/B' ratio (L' and B' -effective foundation dimensions), width of the soil layer, Poisson's coefficient ν and depth of the foundation D. m -number of corners contributing to settlement (at a corner of the footing m=1; at the centre m=4).
proposed another equation to obtain ks (method No. 5): (6) where: IF -influence factor, which depends on the ratio D/L and L/B (L-foundation length; Bfoundation width; D -foundation slab depth) and on Poisson's ratio ν; IS -influence factor, which depends on L'/B' ratio (L' and B' -effective foundation dimensions), width of the soil layer, Poisson's coefficient ν and depth of the foundation D. m -number of corners contributing to settlement (at a corner of the footing m=1; at the centre m=4).
proposed another equation to obtain ks (method No. 5): (6) where: IF -influence factor, which depends on the ratio D/L and L/B (L-foundation length; Bfoundation width; D -foundation slab depth) and on Poisson's ratio ν; IS -influence factor, which depends on L'/B' ratio (L' and B' -effective foundation dimensions), width of the soil layer, Poisson's coefficient ν and depth of the foundation D. m -number of corners contributing to settlement (at a corner of the footing m=1; at the centre m=4).
influence factor, which depends on the ratio D/L and L/B (L-foundation length; B-foundation width; D -foundation slab depth) and on Poisson's ratio ν; I S -influence factor, which depends on L'/B' ratio (L' and B' -effective foundation dimensions), width of the soil layer, Poisson's coefficient ν and depth of the foundation D.m -number of corners contributing to settlement (at a corner of the footing m=1; at the centre m=4).
-influence factor, which depends on the ratio D/L and L/B (L-foundation length; Bfoundation width; D -foundation slab depth) and on Poisson's ratio ν; IS -influence factor, which depends on L'/B' ratio (L' and B' -effective foundation dimensions), width of the soil layer, Poisson's coefficient ν and depth of the foundation D. m -number of corners contributing to settlement (at a corner of the footing m=1; at the centre m=4).
half of the length of the foundation; B 1 -half of the foundation width; z i -upper depth of the considered layer.

Fig. 3 .
Fig. 3. Method comparison with No.1 (Winkler's) method for one-layered A and B geologies (left diagram) and for multi-layered C and D geologies (right diagram).
calculated for each one of the soils and for each one of loading cases (500 kN, 1000 kN and 2000 kN), applying the different k s calculation methods.The unified Δ value for each soil is obtained calculating average of the obtained values in each loading case.Finally average Δ value for one-layered soils (A and B) and average Δ value for multi-layered soils (C and D) were obtained.The comparison analysis of these final Δ results is presented in Fig.5.

Fig. 3
Fig. 3 Method comparison with No.1 (Winkler's) method for one-layered A and B geologies (left diagram) and for multi-layered C and D geologies (right diagram)

Fig. 3 .
Fig. 3. Method comparison with No.1 (Winkler's) method for one-layered A and B geologies (left diagram) and for multi-layered C and D geologies (right diagram).
Hsl -layer thickness; qc -cone tip resistance; E0 -deformation modulus; c' -effective cohesion; φ -soil friction angle; γ -weight; ν -Poisson's coefficient.The bending moment diagrams under the three different described situations are shown in Fig. 5. Note: boxes of internal forces of columns have solid lines.Boxes of internal forces of beams have dashed lines.Values of bending moments in Fig. 5 are divided by 527,67 in beams and by 161,7 in columns (maximum values).

Fig. 5 .
Fig. 5. Diagrams of bending moments to 21 %.There are also significant differences of bending moments in the columns.Bending moments increased by up to 41 % at the top of the columns C1 and C6.Introducing different types of soil (situation No.

Table 1
Properties of the layers of the soil

Table 3 .
Legend of the diagrams