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Evaluation of ground deformability with respect to vehicle mobility
Journal o f Terramechanics, 1973, Vol. 10, No. 1, pp. 105 to 111. Pergamon Press
Printed in Great Britain.
EVALUATION OF G R O U N D DEFORMABILITY WITH RESPECT TO VEHICLE MOBILITY* YA. S. AGEYKINt To increase vehicle mobility, it is necessary to consider the mechanical properties of ground surfaces, in particular their deformability. In a theoretical discussion of problems of mobility, ground deformability is evaluated by means of the exponential formula: q -- ch~
(l)
where q = specific ground pressure in kg/cm2; h = soil deformation in cm; c and ~ = ground parameters. The broad application of formula (1) is due to its simplicity and universality. It is suitable for description of relationships of various forms. However, this formula possesses shortcomings which are at the present time causing serious difficulties in the development of mobility theory. Parameters c and ~ are assumed to remain constant for a given soil, whereas in reality they vary depending on the amount of surface loading and range of loads. In Fig. 1 the experimentally derived plots of q = F(h) are presented, which are typical for the majority of ground surfaces. Curve 1 is characteristic for soils without a contiguous rigid substratum. To describe this curve by means of formula (1) the parameters ~ and c must be selected for a definite range of loads. For 0 < q < 1 kg/cm 2, ~ _~ 1 and c - 0.3. I f 0 < q < 1.4 kg/cm 2, formula (1) is suitable for only a rough approximation (~ -~ 0.3, c -~ 0.6). Curve 2 is characteristic for frequently encountered ground surfaces consisting of a relatively thin, "soft" upper layer and a more resistant and rigid lower layer. It is impossible to closely describe this relationship using an exponential formula. Only separate parts of this curve can be obtained with formula (1). Values of c and ~ will vary within very broad limits. In Fig. 2 the most characteristic relationship between depth of depression of a footing and its diameter at constant unit load is shown. Similar relationships for h = F ( D ) with a clearly expressed minimum have been obtained by Puzakev, Koshavnzy, Gaponenko et al. In formula (1) the effect of footing dimensions on soil depression is not taken into account. Therefore, when formula (1) is used to describe depression curves of differing footings in the same soil, it is necessary to choose different values for e and ~t. *Translated from Automobil 'naya Promyshlennost, No. 6 (1970) by J. McVay; contributed by Z. J. Janosi. 105
106
YA. S. A G E Y K I N 1
I
I
//I/
%
0'8
I//
/ q
,."
1.2
,j
I
04
I
I
I
I
4
8
12
16
h,
FIG. 1.
1 20
Cm
Characteristics of penetrometer sinkage.
Parameters c and Exhave no physical significance. They are empirical constants for specific, narrowly defined conditions of footing depression in soil. The dimensionality of c varies with changes in the value of ~z. All this introduces serious difficulties when formula (1) is employed. A more theoretically sound formula is: q ---- c
,
(2)
where D is the diameter of a circle whose area is equal to that of the press tool. In formula (2) parameter c has a constant dimension (kg/cm2). The effect of penetrometer footing dimensions on soil depression is taken into account, although not completely. The function h ---- F(D), where q ---- const in accordance with formula (2) is linear, passing through the origin of coordinates. Comparison of this relationship I
I
I /
/ I0
/
/
/
/
/ / / \
"-.
./
i
i
5
I0
/
D,
FIG. 2.
"--t-L_ 15
I
_ 20
25
cm
Relationships of penetrometer sinkage and footing diameter for q ~ 0.5 kg/cm 2. 1. Loam; 2. Sand.
EVALUATION OF GROUND DEFORMABILITY
107
with experimentally derived relationships (Fig. 2) demonstrates the limited possibilities of formula (2) for describing the experimental curves. In recent years, foreign investigators have applied the following modified formula proposed by Bekker [1]:
q=(cl
+~2-)h~
(3)
where b is footing width; and Cl, cg and ~t are soil parameters.
In contrast to the foregoing, formula (3) takes into account the effect of footing dimensions in soil deformation for various soils by means of a different relationship between cl and c2. However, even this formula corresponds to experimental data and to the physical process of soil deformation only for limited conditions. In formula (3) the effect of penetrometer footing dimensions are considered unilaterally. Taken into account is the increase with greater footing dimensions of deformations occurring as a result of the propagation of soil deformations to a greater depth. Not taken into account, however, is the decrease in relative side shear of soil with greater footing dimensions as a result of which overall deformation decreases. Therefore, the nature of the relationships computed in accordance with formula (3) is close to experimental results only in those instances where shear information is in little evidence (in Fig. 2--the right portions of the curves). The left portions of the experimentally derived curves, which characterize a decrease in footing diameter and an increase in soil deformation as a result of greater shear deformations, do not agree with the relationships obtained by means of formula (3). The dimensionality of parameters Cl and c~ is variable, it varies according to the values of ~t. Thus, none of the three applied exponential formulas reflects to the required degree the effect of loading-surface dimensions on soil deformation. In addition, it is impossible to describe with exponential formulas the relationship between deformation and load for soils with a continguous hard substratum. The author presents a new formula in the following. When a footing acts on soil, two types of deformation are produced: compression of soil particles and soil shear.
; ~ - ~'.
(4)
E
The relationship between relative comprressive deformation 2 and stresses ~ may be assumed to be expressed linearly as in equation 4, where E is the soil deformation coefficient in the absence of shear. To determine the absolute compressive deformation it is necessary to know the stress distribution in soil with respect to depth, as well as the thickness of the deformed soil layer. We will express the stress distribution in soil by means of the equation: --
q
l+
(5)
~-D
108
YA.
S. A G E Y K I N
where z - - distance f r o m the observed f u n d a m e n t a l soil layer to the f o o t i n g soil contact surface: d =- d i a m e t e r o f a circle equal to p e n e t r o m e t e r area; a coefficient characterizing a t t e n u a t i o n o f stresses in soil. E q u a t i o n (5) is s u b s t a n t i a t e d by the e x p e r i m e n t a l d a t a o f U l ' y a n o v [2]. C o m p u t a t i o n o f compressive d e f o r m a t i o n is illustrated in Fig. 3. "////,,,7~52v///g,x\\~ q
•~\v@,x\\\\ -~/~\\x\\ "~/\\ cr\ \~ 3 \\\ \'~/\X\\ ~
.
dz
7/1IIIIII//1IIIIIII//111111 Z
7/7///////,
//
FI(;. 3. Sketch to illustrate soil reformation. W e shall express the vertical compression o f a f u n d a m e n t a l layer with initial thickness dz in terms o f the relative d e f o r m a t i o n . dh =: X dz o.
(6)
Thickness o f the f u n d a m e n t a l layer in a d e f o r m e d state is: dz == dz 0 (1--)~) .
(7)
Solving equations (4), (6) and (7) simultaneously we o b t a i n : ¢;dz -
dh - -
.
(S)
E--r~
I n a c c o r d a n c e with the d a t a o f l v a n o v [3], q J E = 0.0125 to 0"003; qs is the l o a d - b e a r i n g capacity o f the soil. C o n s i d e r i n g that ~ ~
~-
qdz
..
(9) Z
EVALUATION OF GROUND DEFORMABILITY
109
After integration of this equation between the limits z = 0, z = ( H -- h) we obtain the expression for compressive deformation of soil: h = qaD arc tan H - --- h E aD
(10)
The relative portion of deformations due to shear increases progressively with increase in load. In the work by Ornatskiy [4] an increase in overall deformation due to soil shear is taken into account by multiplying the compressive deformation by the quantity q~/(q~- q). This expression is used for evaluation of deformation of the fundamental volume of soil. However, as many investigators indicate, the entire compressed core undergoes a downward displacement as a result of shear deformation. Therefore, it is more advantageous to apply the expression qJ(q~ -- q) to evaluate the relative shear distortion of the entire soil volume. Multiplying this expression by the value of compressive deformation (10) we obtain an equation for the relationship between load and total soil deformation: 1
q = 1
aD
qs
Eh
- + --
(11) H--h
arc tan - -
aD
Several formulas are known in soil mechanics for determination of soil load-bearing capacity q~. In comparing these formulas Maslov [5] came to the conclusion that results obtained by application of the various formulas differ to an insignificant degree. The majority of proposed formulas reduce to the expression: (12)
q s = X I D + X2 + X3h
where XI, X~ and X a are parameters expressed in terms of soil unit weight 7, angle of internal friction % and soil cohesion co. Values of )(1, X2 and )(3 are presented in the work by Terzaghi [6]. In equation (12) the effect of a contiguous hard layer is not considered. With the approach of the footing to the hard substratum, the load-bearing capacity of the soil increases as a result of decrease in lateral forces. This increase in soil load-bearing capacity may be expressed by the equation: q~ =
n qs 2 a r c tan(-~hDh )
(13)
where qs is the load-bearing capacity of soil in the absence of a contiguous hard substratum as expressed by equation (12). Substituting q~ for qs in equation (11) we obtain computation formula: 1
q =
2
--arctan-nqs
H--h aD
aD
H--h"
Eh
aD
+ - - arc tan - -
(14)
110
YA. $. A G E Y K I N
In contrast to the exponential formulas (1), (2) and (3), formula (14) permits a more accurate description of relationships which are characteristic for a majority of soils, including soils with a contiguous hard substratum. This formula more completely takes into account the effect of footing dimensions on soil deformation and gives a sufficiently close description of the complex function h ~ F(d) (Fig. 2). Formula (14) may be simplified to solve many specific problems. In the absence of a contiguous hard layer, the following may be assumed: H--h
H ~ ~,arctan
7t
--
aD
-
2
then q
z
.
.
.
.
1
.
(15)
.
1
naD
q~
2Eh
The expression for q~ may also be simplified: for sandy soils with l ' , x 0, (16)
q, = X I D + X,h .
For clayey soils, X1 -~ 01t Xs __- 0 q~ -~ ,~2 •
(17)
In the case of water-saturated soils without a contiguous hard substratum, soil deformation is basically a result of side shear, while compressive deformation is very slight. For these conditions, it may be assumed that (aD/Eh) ~ 0 since the value is small compared to the first term 1/q, : q - - - - q s = X1D + X , + X , h .
(18)
For dense homogeneous soils it may be assumed that: 1
H--h
-
-~ 0 , a r c t a n -
q~
Eh
~
~ ,~, q . . . . .
aD
(19)
7taD
For ground with a thin layer of soft soil: arc tan
H--h aD
~ H--h, 1 ~ 0 aD
q~
therefore: q --
Eh H--h
(20)
EVALUATION OF GROUND DEFORMABILITY
111
Equation (20) is similar to the equation obtained by B a b k o v [7] for a thin layer of deformed soil. Thus, for solving the majority o f specific problems, formula (14) is substantially simplified and is acceptable for computations without the use o f computers.
REFERENCES [1] M.G. BEKKER,Off-the-RoadLocomotion. The University of Michigan Press (1960). [2] N.A. UL'YANOV,Principles of the Theory and Design of Wheeled Running Gear of Earth-Digging Machinery.Mashgiz (1962). [3] N.N. IVANOV,Basic Aspects of Soil Mechanics Affecting Mobility. Academy of Sciences of the USSR (1950). [4] I.V. ORNATSKIY,Soil Mechanics. Moscow State University (1950). [5] N.N. MASLOV.Applied Soil Mechanics. Mashstroyizdat (1949). [6] K. TERZAGHI,Theory of Soil Mechanics. Gosstroyizdat (1961). [7] V. F. BABKOV, A. K. BIRULYAand V. M. SIDENKO,Ground Mobility of Wheeled Vehicles. Avtotransizdat (1959).
Printed in Great Britain.
EVALUATION OF G R O U N D DEFORMABILITY WITH RESPECT TO VEHICLE MOBILITY* YA. S. AGEYKINt To increase vehicle mobility, it is necessary to consider the mechanical properties of ground surfaces, in particular their deformability. In a theoretical discussion of problems of mobility, ground deformability is evaluated by means of the exponential formula: q -- ch~
(l)
where q = specific ground pressure in kg/cm2; h = soil deformation in cm; c and ~ = ground parameters. The broad application of formula (1) is due to its simplicity and universality. It is suitable for description of relationships of various forms. However, this formula possesses shortcomings which are at the present time causing serious difficulties in the development of mobility theory. Parameters c and ~ are assumed to remain constant for a given soil, whereas in reality they vary depending on the amount of surface loading and range of loads. In Fig. 1 the experimentally derived plots of q = F(h) are presented, which are typical for the majority of ground surfaces. Curve 1 is characteristic for soils without a contiguous rigid substratum. To describe this curve by means of formula (1) the parameters ~ and c must be selected for a definite range of loads. For 0 < q < 1 kg/cm 2, ~ _~ 1 and c - 0.3. I f 0 < q < 1.4 kg/cm 2, formula (1) is suitable for only a rough approximation (~ -~ 0.3, c -~ 0.6). Curve 2 is characteristic for frequently encountered ground surfaces consisting of a relatively thin, "soft" upper layer and a more resistant and rigid lower layer. It is impossible to closely describe this relationship using an exponential formula. Only separate parts of this curve can be obtained with formula (1). Values of c and ~ will vary within very broad limits. In Fig. 2 the most characteristic relationship between depth of depression of a footing and its diameter at constant unit load is shown. Similar relationships for h = F ( D ) with a clearly expressed minimum have been obtained by Puzakev, Koshavnzy, Gaponenko et al. In formula (1) the effect of footing dimensions on soil depression is not taken into account. Therefore, when formula (1) is used to describe depression curves of differing footings in the same soil, it is necessary to choose different values for e and ~t. *Translated from Automobil 'naya Promyshlennost, No. 6 (1970) by J. McVay; contributed by Z. J. Janosi. 105
106
YA. S. A G E Y K I N 1
I
I
//I/
%
0'8
I//
/ q
,."
1.2
,j
I
04
I
I
I
I
4
8
12
16
h,
FIG. 1.
1 20
Cm
Characteristics of penetrometer sinkage.
Parameters c and Exhave no physical significance. They are empirical constants for specific, narrowly defined conditions of footing depression in soil. The dimensionality of c varies with changes in the value of ~z. All this introduces serious difficulties when formula (1) is employed. A more theoretically sound formula is: q ---- c
,
(2)
where D is the diameter of a circle whose area is equal to that of the press tool. In formula (2) parameter c has a constant dimension (kg/cm2). The effect of penetrometer footing dimensions on soil depression is taken into account, although not completely. The function h ---- F(D), where q ---- const in accordance with formula (2) is linear, passing through the origin of coordinates. Comparison of this relationship I
I
I /
/ I0
/
/
/
/
/ / / \
"-.
./
i
i
5
I0
/
D,
FIG. 2.
"--t-L_ 15
I
_ 20
25
cm
Relationships of penetrometer sinkage and footing diameter for q ~ 0.5 kg/cm 2. 1. Loam; 2. Sand.
EVALUATION OF GROUND DEFORMABILITY
107
with experimentally derived relationships (Fig. 2) demonstrates the limited possibilities of formula (2) for describing the experimental curves. In recent years, foreign investigators have applied the following modified formula proposed by Bekker [1]:
q=(cl
+~2-)h~
(3)
where b is footing width; and Cl, cg and ~t are soil parameters.
In contrast to the foregoing, formula (3) takes into account the effect of footing dimensions in soil deformation for various soils by means of a different relationship between cl and c2. However, even this formula corresponds to experimental data and to the physical process of soil deformation only for limited conditions. In formula (3) the effect of penetrometer footing dimensions are considered unilaterally. Taken into account is the increase with greater footing dimensions of deformations occurring as a result of the propagation of soil deformations to a greater depth. Not taken into account, however, is the decrease in relative side shear of soil with greater footing dimensions as a result of which overall deformation decreases. Therefore, the nature of the relationships computed in accordance with formula (3) is close to experimental results only in those instances where shear information is in little evidence (in Fig. 2--the right portions of the curves). The left portions of the experimentally derived curves, which characterize a decrease in footing diameter and an increase in soil deformation as a result of greater shear deformations, do not agree with the relationships obtained by means of formula (3). The dimensionality of parameters Cl and c~ is variable, it varies according to the values of ~t. Thus, none of the three applied exponential formulas reflects to the required degree the effect of loading-surface dimensions on soil deformation. In addition, it is impossible to describe with exponential formulas the relationship between deformation and load for soils with a continguous hard substratum. The author presents a new formula in the following. When a footing acts on soil, two types of deformation are produced: compression of soil particles and soil shear.
; ~ - ~'.
(4)
E
The relationship between relative comprressive deformation 2 and stresses ~ may be assumed to be expressed linearly as in equation 4, where E is the soil deformation coefficient in the absence of shear. To determine the absolute compressive deformation it is necessary to know the stress distribution in soil with respect to depth, as well as the thickness of the deformed soil layer. We will express the stress distribution in soil by means of the equation: --
q
l+
(5)
~-D
108
YA.
S. A G E Y K I N
where z - - distance f r o m the observed f u n d a m e n t a l soil layer to the f o o t i n g soil contact surface: d =- d i a m e t e r o f a circle equal to p e n e t r o m e t e r area; a coefficient characterizing a t t e n u a t i o n o f stresses in soil. E q u a t i o n (5) is s u b s t a n t i a t e d by the e x p e r i m e n t a l d a t a o f U l ' y a n o v [2]. C o m p u t a t i o n o f compressive d e f o r m a t i o n is illustrated in Fig. 3. "////,,,7~52v///g,x\\~ q
•~\v@,x\\\\ -~/~\\x\\ "~/\\ cr\ \~ 3 \\\ \'~/\X\\ ~
.
dz
7/1IIIIII//1IIIIIII//111111 Z
7/7///////,
//
FI(;. 3. Sketch to illustrate soil reformation. W e shall express the vertical compression o f a f u n d a m e n t a l layer with initial thickness dz in terms o f the relative d e f o r m a t i o n . dh =: X dz o.
(6)
Thickness o f the f u n d a m e n t a l layer in a d e f o r m e d state is: dz == dz 0 (1--)~) .
(7)
Solving equations (4), (6) and (7) simultaneously we o b t a i n : ¢;dz -
dh - -
.
(S)
E--r~
I n a c c o r d a n c e with the d a t a o f l v a n o v [3], q J E = 0.0125 to 0"003; qs is the l o a d - b e a r i n g capacity o f the soil. C o n s i d e r i n g that ~ ~
~-
qdz
..
(9) Z
EVALUATION OF GROUND DEFORMABILITY
109
After integration of this equation between the limits z = 0, z = ( H -- h) we obtain the expression for compressive deformation of soil: h = qaD arc tan H - --- h E aD
(10)
The relative portion of deformations due to shear increases progressively with increase in load. In the work by Ornatskiy [4] an increase in overall deformation due to soil shear is taken into account by multiplying the compressive deformation by the quantity q~/(q~- q). This expression is used for evaluation of deformation of the fundamental volume of soil. However, as many investigators indicate, the entire compressed core undergoes a downward displacement as a result of shear deformation. Therefore, it is more advantageous to apply the expression qJ(q~ -- q) to evaluate the relative shear distortion of the entire soil volume. Multiplying this expression by the value of compressive deformation (10) we obtain an equation for the relationship between load and total soil deformation: 1
q = 1
aD
qs
Eh
- + --
(11) H--h
arc tan - -
aD
Several formulas are known in soil mechanics for determination of soil load-bearing capacity q~. In comparing these formulas Maslov [5] came to the conclusion that results obtained by application of the various formulas differ to an insignificant degree. The majority of proposed formulas reduce to the expression: (12)
q s = X I D + X2 + X3h
where XI, X~ and X a are parameters expressed in terms of soil unit weight 7, angle of internal friction % and soil cohesion co. Values of )(1, X2 and )(3 are presented in the work by Terzaghi [6]. In equation (12) the effect of a contiguous hard layer is not considered. With the approach of the footing to the hard substratum, the load-bearing capacity of the soil increases as a result of decrease in lateral forces. This increase in soil load-bearing capacity may be expressed by the equation: q~ =
n qs 2 a r c tan(-~hDh )
(13)
where qs is the load-bearing capacity of soil in the absence of a contiguous hard substratum as expressed by equation (12). Substituting q~ for qs in equation (11) we obtain computation formula: 1
q =
2
--arctan-nqs
H--h aD
aD
H--h"
Eh
aD
+ - - arc tan - -
(14)
110
YA. $. A G E Y K I N
In contrast to the exponential formulas (1), (2) and (3), formula (14) permits a more accurate description of relationships which are characteristic for a majority of soils, including soils with a contiguous hard substratum. This formula more completely takes into account the effect of footing dimensions on soil deformation and gives a sufficiently close description of the complex function h ~ F(d) (Fig. 2). Formula (14) may be simplified to solve many specific problems. In the absence of a contiguous hard layer, the following may be assumed: H--h
H ~ ~,arctan
7t
--
aD
-
2
then q
z
.
.
.
.
1
.
(15)
.
1
naD
q~
2Eh
The expression for q~ may also be simplified: for sandy soils with l ' , x 0, (16)
q, = X I D + X,h .
For clayey soils, X1 -~ 01t Xs __- 0 q~ -~ ,~2 •
(17)
In the case of water-saturated soils without a contiguous hard substratum, soil deformation is basically a result of side shear, while compressive deformation is very slight. For these conditions, it may be assumed that (aD/Eh) ~ 0 since the value is small compared to the first term 1/q, : q - - - - q s = X1D + X , + X , h .
(18)
For dense homogeneous soils it may be assumed that: 1
H--h
-
-~ 0 , a r c t a n -
q~
Eh
~
~ ,~, q . . . . .
aD
(19)
7taD
For ground with a thin layer of soft soil: arc tan
H--h aD
~ H--h, 1 ~ 0 aD
q~
therefore: q --
Eh H--h
(20)
EVALUATION OF GROUND DEFORMABILITY
111
Equation (20) is similar to the equation obtained by B a b k o v [7] for a thin layer of deformed soil. Thus, for solving the majority o f specific problems, formula (14) is substantially simplified and is acceptable for computations without the use o f computers.
REFERENCES [1] M.G. BEKKER,Off-the-RoadLocomotion. The University of Michigan Press (1960). [2] N.A. UL'YANOV,Principles of the Theory and Design of Wheeled Running Gear of Earth-Digging Machinery.Mashgiz (1962). [3] N.N. IVANOV,Basic Aspects of Soil Mechanics Affecting Mobility. Academy of Sciences of the USSR (1950). [4] I.V. ORNATSKIY,Soil Mechanics. Moscow State University (1950). [5] N.N. MASLOV.Applied Soil Mechanics. Mashstroyizdat (1949). [6] K. TERZAGHI,Theory of Soil Mechanics. Gosstroyizdat (1961). [7] V. F. BABKOV, A. K. BIRULYAand V. M. SIDENKO,Ground Mobility of Wheeled Vehicles. Avtotransizdat (1959).