An
analytical method of pore pressure calculations when cutting water saturated
sand.
Dr.ir. S.A. Miedema[1]
Y. Zhao[2]
ABSTRACT
In the cutting of water-saturated sand, the phenomena of dilatation causes the development of pore vacuum pressures. These vacuum pressures result in an increase of the grain stresses, resulting in very high cutting forces. Until now the calculation of the pore vacuum pressure has been a matter of Finite Element Calculations.
The Finite Element Calculations have resulted in dimensionless coefficient tables. To use these tables (as published by Miedema), one had to use interpolation or extrapolation methods. This is all very time consuming and it makes the use difficult and hard to understand.
By using an electrical analogon for the pore vacuum pressures, almost the same results can be obtained by using the theory of parallel resistors. Only a few lines of programming code are required.
The FEM calculations have been carried out again to get a correct reference. It appeared that the results of the FEM calculations have not changed much. The difference between the calculations carried out by Miedema (1987 [23]) and the calculations carried out by Zhao (2000, [30]) is less then 1%. The results of the analytical method differ more, within 10% for the relevant cases, depending on the blade geometry, but this is still very good for a first estimate of the pore vacuum pressures.
The paper will first show the state of the art Finite Element results and then the electrical analogon. The results will be compared.
Keywords: Dredging, soil mechanics, cutting theories
INTRODUCTION
One of the strong non-linear
effects in the equilibrium equations of motion for the determination of the
cutter-suction and dredging-wheel dredger motions, is the interaction between
the excavating element and the soil. A good description of the cutting process
is essential for a reliable simulation of the ship motions, in order to be able
to predict the usability and the design of sea-going dredging vessels.
Although calculation models for the determination of the cutting forces
for dry sand were available for a long time (Hettiaratchi & Reece
1966-1975, [3, 4, 5, 6, 7, 26], Hatamura & Chiiwa 1975-1977, [2] etc.) it
is only since the seventies and the eighties that the cutting process in
saturated sand is extensively researched at the Delft Hydraulics in Delft (WL,
CSB), at the Delft University of Technology and at the Mineral Technological
Institute (MTI, IHC). First the process is described, for a good understanding
of the terminology used in the literature discussion. From literature it is
known that, during the cutting process, the sand increases in volume (see
figure 1). This increase in volume is accredited to dilatancy. This is the
change of the pore volume as a result of shear in the sand package. This
increase of the pore volume has to be filled with water. The flowing water
experiences a certain resistance, which causes vacuum pressure in the pore
water in the sand package. As a result the grain stresses increase and
therefore the needed cutting forces. The speed of the increase of the pore
volume in the dilatancy zone, the volume strain rate, is proportional to the
cutting velocity. If the volume strain rate is high, there is a chance that the
pore pressure reaches the saturated water vapor pressure and cavitation occurs.
A further increasing volume strain rate will not be able to cause a further
decrease of the pore pressure. This also implies that, with a further
increasing cutting velocity, the cutting forces cannot increase as a result of
the dilatancy properties of the sand. The cutting forces can, however, still
increase with an increasing cutting velocity as a result of the inertia forces
and the flow resistance.
The cutting process can be subdivided in 5 areas in relation with the
cutting forces:
- Very low cutting velocities, a quasi
static cutting process. The cutting forces are determined by the
gravitation, cohesion and adhesion.
- The volume strain rate is high in
relation to the permeability of the sand. The volume strain rate is
however so small that inertia forces can be neglected. The cutting forces
are dominated by the dilatancy properties of the sand.
- A transition region, with local
cavitation. With an increasing volume strain rate, the cavitation area
will increase so that the cutting forces increase slightly as a result of
dilatancy.
- Cavitation occurs almost everywhere around
and on the blade. The cutting forces do not increase anymore as a result
of the dilatancy properties of the sand.
- Very high cutting velocities. The
inertia forces part in the total cutting forces can no longer be neglected
but form a substantial part.
Cutting theory,
literature survey
In the seventies extensive research is carried out on the forces that occur while cutting sand under water. A conclusive cutting theory has however not been published in this period. However qualitative relations have been derived by several researchers, with which the dependability of the cutting forces with the soil properties and the blade geometry are described (Joanknecht 1974, [9], van Os 1977, [24, 25]).
Afterwards
it turned out that, in non-published reports for the confidential research
program CSB, as indicated in the reference list of [12], Van Os had already
developed the basic theory for the cutting of saturated packed sand. Ahead of
the real publication, [12] is provided by the Delft Hydraulics Laboratory in
August 1986.
A process that has a lot of similarities with the cutting of sand as far as water pressure development is concerned, is the, with uniform velocity, forward moving breach. Meijer and van Os (1976, [13]) and Meijer (1981-1985, [14, 15]) have transformed the storage equation for the, with the breach, forward moving coordinate system.
|
|
(1) |
In case of a stationary process, the second term on the right hand side is zero, resulting:
|
|
(2) |
Van Os (1977, [24, 25]) describes the basic principles of the cutting process, with special attention for the determination of the water vacuum pressures and the cavitation. Van Os uses the non-transformed storage equation for the determination of the water vacuum pressures.
|
|
(3) |
Figure 3: The volume balance over the shear zone.
The average volume strain rate has to be substituted in the term e/t on the right hand side. The average volume strain rate is the product of the average volume strain of the sand package and the cutting velocity and arises from the volume balance over the shear zone. Van Os gives a qualitative relation between the water vacuum pressures and the average volume strain rate:
|
|
(4) |
The problem of the solution of the storage equation for the cutting of sand under water is a mixed boundary-value problem, for which the water vacuum pressures along the boundaries are known (hydrostatic).
Joanknecht (1973, [8, 9]) assumes that the cutting forces are determined by the vacuum pressure in the sand package. A distinction is made between the parts of the cutting force caused by the inertia forces, the vacuum pressure behind the blade and the soil mechanical properties of the sand. The influence of the geometrical parameters gives the following qualitative relation:
|
|
(5) |
The cutting force is proportional to the cutting velocity, the blade width and the square of the initial layer-thickness. A relation with the pore percentage and the permeability is also mentioned. A relation between the cutting force and these soil mechanical properties is however not given. It is observed that the cutting forces increase with an increasing blade angle.
In the eighties research has led to more quantitative relations. In 1984 Van Leussen and Nieuwenhuis [11] discuss the soil mechanical aspects of the cutting process. The forces models of Miedema [20, 23] and Steeghs [28, 29]) are published 1985/86, while the CSB (Combinatie Speurwerk Baggertechniek) model (van Leussen en van Os [12]), however developed in the early seventies at the Delft Hydraulics Laboratory (van Os [24, 25]), has been published in 1987. Brakel (1982, [1]) derives a relation for the determination of the water vacuum pressures based upon, over each other rolling, round grains in the shear zone. The force part resulting from this is added to the model of Hettiaratchi and Reece (1974, [6]). Miedema (1983, [19]) has combined the qualitative relations of Joanknecht (1973, [8, 9]) and Van Os (1977, [24, 25]) to the following relation:
|
|
(6) |
With this basic equation calculation models are developed for a cutter head and for the periodical moving cutter head in the breach. The proportionality constants are determined empirically.
Figure 4: The flow of the pore water towards the shear zone.
Van Leussen and Nieuwenhuis (1984, [11]) discuss the soil mechanical aspects of the cutting process. Important in the cutting process is the way shear takes place and the shape or angle of the shear plane, respectively shear zone. In literature no unambiguous image could be found. Cutting tests along a windowpane gave an image in which the shape of the shear plane was more in accordance with the so-called "stress characteristics" than with the so-called "zero-extension lines". Therefore, for the calculation of the cutting forces, the "stress characteristics method" is used (Mohr-Coulomb failure criterion). For the calculation of the water vacuum pressures, however, the "zero-extension lines" are used, which are lines with a zero linear strain. A closer description has not been given for both calculations. Although the cutting process is considered as being two-dimensional, Van Leussen and Nieuwenhuis found, that the angle of internal friction, measured at low deformation rates in a triaxial apparatus, proved to be sufficient for dredging processes. Although the cutting process can be considered as a two-dimensional process and therefore it should be expected that the angle of internal friction has to be determined with a "plane deformation test". A sufficient explanation has not been found.
Little is known about the value of the angle of friction between sand and steel. Van Leussen and Nieuwenhuis don't give an unambiguous method to determine this soil mechanical parameter. It is, however, remarked that at low cutting velocities (0.05 mm/s), the soil/steel angle of friction can have a statistical value, which is 1.5 to 2 times larger than the dynamic soil/steel angle of friction. The influence of the initial density on the resulting angle of friction is not clearly present, because loose packed sand moves over the blade. The angles of friction measured on the blades are much larger than the angles of friction measured with an adhesion cell, while also a dependency with the blade angle is observed.
With regard to the permeability of the sand, Van Leussen and Nieuwenhuis found that no large deviations of Darcy's law occur with the water flow through the pores. The found deviations are in general smaller than the accuracy with which the permeability can be determined in situ.
The size of the area where e/t from equation (5) is zero can be clarified by the figures published by Van Leussen and Nieuwenhuis. The basis is formed by a cutting process where the density of the sand is increased in a shear band with a certain width. The undisturbed sand has the initial density while the sand after passage of the shear band possesses a critical density. This critical density appeared to be in good accordance with the wet critical density of the used types of sand. This implies that outside the shear band the following equation is valid:
|
|
(7) |
Values for the various densities are given for three types of sand. Differentiation of the residual density as a function of the blade angle is not given. A verification of the water pressure calculations is given for a 60 blade with a blade-height/layer-thickness ratio of 1.
Figure 5: The course mesh as applied in the pore vacuum pressure calculations.
Miedema (1984, [18, 19]) gives a formulation for the determination of the water vacuum pressures. The deformation rate is determined by taking the volume balance over the shear zone, as Van Os (1977, [24, 25]) did. The deformation rate is modeled as a boundary condition in the shear zone, while the shear zone is modeled as a straight line instead of a shear band as with Van Os (1977, [24, 25]) and Van Leussen and Nieuwenhuis (1987, [11]). The influence of the water depth on the cutting forces is clarified, as is shown in figure 3.
Steeghs (1985, [28]) developed a theory for the determination of the volume strain rate, based upon a cyclic deformation of the sand in a shear band. This implies that not an average value is taken for the volume strain rate but a cyclic, with time varying, value, based upon the dilatancy angle theory.
Miedema (1985, [20, 21]) derives equations for the determination of the water under-pressures and the cutting forces, based upon [16, 18, 19]. The water vacuum pressures are determined with a finite element method. Explained are the influence of the permeability of the disturbed and undisturbed sand and the determination of the shear angle. The derived theory is verified with model tests. On basis of this research nmax is chosen for the residual pore percentage instead of the wet critical density.
Figure 6: The fine mesh as applied in the pore vacuum pressure calculations.
Steeghs (1985-1986, [28, 29]) derives equations for the determination of the water vacuum pressures according an analytical approximation method. With this approximation method the water vacuum pressures are determined with a modification of equation (4) derived by Van Os (1977,[24, 25]) and the storage equation (7). Explained is how cutting forces can be determined with the force equilibrium on the cut layer. Also included are the gravity force, the inertia forces and the vacuum pressure behind the blade. For the last influence factor no formulation is given. Discussed is the determination of the shear angle. Some examples of the cutting forces are given as a function of the cutting velocity, the water depth and the sub-pressure behind the blade. A verification of this theory is not given.
Miedema (1986, [22]) develops a calculation model for the determination of the cutting forces on a dredging-wheel based upon [20, 21]. Also nomograms are published with which the cutting forces and the shear angle can be determined in a simple way. Explained is the determination of the weighted average permeability from the permeability of the disturbed and undisturbed sand. Based upon the calculations it is concluded that the average permeability forms a good estimation.
Miedema (1986, [23]) extends the theory with adhesion, cohesion, inertia forces, gravity, and vacuum pressure behind the blade. The method for the calculation of the coefficients for the determination of a weighed average permeability are discussed. It is concluded that the additions to the theory lead to a better correlation with the tests results.
Van Os and Van Leussen (1986, [12]) summarize the publications of Van Os (1977, [24, 25]) and of Van Leussen and Nieuwenhuis (1987, [11]) and give a formulation of the theory developed in the early seventies at the Delft Hydraulics Laboratory. Discussed are the water pressures calculation, cavitation, the weighed average permeability, the angle of internal friction, the soil/steel angle of friction, the permeability, the volume strain and the cutting forces. Verification is given of a water pressures calculation and the cutting forces. The water vacuum pressures are determined with equation (4) derived by Van Os (1977, [24, 25]). The water pore pressure calculation is performed with the finite difference method, in which the height of the shear band is equal to the mesh width of the grid. The size of this mesh width is considered to be arbitrary. From an example, however, it can be seen that the shear band has a width of 13% of the layer-thickness. Discussed is the determination of a weighed average permeability. The forces are determined with Coulomb's method.
Determination of the
pore under-pressure around the blade
The cutting process can be modeled as a two-dimensional process, in which a straight blade cuts a small layer of sand (figure 1). The sand is deformed in the shear zone, also called deformation zone or dilatancy zone. During this deformation the volume of the sand changes as a result of the shear stresses in the shear zone. In soil mechanics this phenomenon is called dilatancy. In densely packed sand the pore volume is increased as a result of the shear stresses in the deformation zone. This increase in the pore volume is thought to be concentrated in the deformation zone, with the deformation zone modeled as a straight line (line sink). Water has to flow to the deformation zone to fill up the increase of the pore volume in this zone. As a result of this water flow the grain stresses increase and the water pressures decrease. Therefore there are water vacuum pressures. This implies that the forces necessary for cutting densely packed sand under water will be determined for an important part by the dilatancy properties of the sand. At low cutting velocities these cutting forces are also determined by the gravity, the cohesion and the adhesion for as far as these last two soil mechanical parameters are present in the sand. Is the cutting carried out at high velocities, than the inertia forces will have an important part in the total cutting forces.
Figure 7: The distribution of the pore vacuum pressure in the sand around the blade.
If the cutting process is assumed
to be stationary, the water flow through the pores of the sand can be described
in a blade motions related coordinate system. The determination of the water
vacuum pressures in the sand around the blade is then limited to a mixed
boundary conditions problem. The potential theory can be used to solve this
problem. For the determination of the water vacuum pressures it is necessary to
have a proper formulation of the boundary condition in the shear zone. Miedema
(1984, [19]) derived the basic equation for this boundary condition.
In 1985 [20, 21] and 1986 [23] a more extensive derivation is published.
If it is assumed that no
deformations take place outside the deformation zone, then:
|
|
(8) |
applies for the sand package around the blade.
The boundary condition is in fact a specific flow rate (fig. 3) that can be determined with the following hypothesis.
For a sand element in the
deformation zone, the increase in the pore volume per unit of blade length, is:
|
|
(9) |
|
In which: |
|
(10) |
For the residual pore percentage is chosen for nmax on the basis of the ability to explain the water vacuum pressures, measured in the laboratory tests.
Figure 8: The distribution of the pore vacuum pressure in the sand around the blade.
The volume flow rate flowing to the sand element, is equal
to:
|
|
(11) |
With the aid of Darcy's law the
next differential equation can be derived for the specific flow rate,
perpendicular to the deformation zone:
|
|
(12) |
The partial derivative p/n is the derivative of the water vacuum pressures perpendicular on the boundary of the area, in which the water vacuum pressures are calculated (in this case the deformation zone). The boundary conditions on the other boundaries of this area are indicated in figure 3. A hydrostatic pressure distribution is assumed on the boundaries between sand and water. This pressure distribution equals zero in the calculation of the water vacuum pressures, if the height difference over the blade is neglected. The boundaries that form the edges in the sand package are assumed to be impermeable.
Making equation (12) dimensionless is similar to that of the breach equation of Meijer and Van Os [13]. In the breach problem the length dimensions are normalized by dividing them by the breach height, while in the cutting of sand they are normalized by dividing them by the cut layer thickness.
Equation (12) in normalized
format:
|
|
(13) |
With: n' = n/hi
Figure 9: The pore vacuum pressure distribution on the blade and on the shear zone.
This equation is made dimensionless with:
|
|
(14) |
The accent indicates that a
certain variable or partial derivative is dimensionless. The next dimensionless
equation is now valid as a boundary condition in the deformation zone:
|
|
(15) |
The storage equation also has to be made dimensionless,
which results in the next equation:
|
|
(16) |
Because the right hand side of this equation equals zero, it is similar to equation (8)
The water vacuum pressures distribution in the sand package can now be determined using the storage equation and the boundary conditions. Because the calculation of the water vacuum pressures is dimensionless the next transformation has to be performed to determine the real water vacuum pressures.
The real water vacuum pressures
can be determined by integrating the derivative of the water vacuum pressures
in the direction of a flow line, along a flow line, so:
|
|
On a streamline s' |
(17) |
This is illustrated in figure 4.
Using equation (14) this can be written as:
|
|
(18) |
With: s' = s/hi
This gives the following relation between the really emerging water
under-pressures and the calculated water under-pressures:
|
|
(19) |
In table 1 the calculated water vacuum pressures are listed in relation with the blade angle, the shear angle, the blade-height/layer-thickness ratio and the ratio between the permeability of the disturbed and undisturbed sand. Using equation (19) or equation (14) also the water vacuum pressures, measured in the cutting tests, can be made dimensionless. To be independent of the ratio between the initial permeability ki and the maximum permeability kmax, kmax has to be replaced with the weighed average permeability km before making the measured water vacuum pressures dimensionless.
Numerical water pore
pressure calculations
The water vacuum pressures in the sand package on and around the blade are numerically determined using the finite element method. A standard program package is used (Segal 2001, [27]). With the in this package, available "subroutines" a program is written, with which water vacuum pressures can be calculated and be output graphically and numerically. The solution of such a calculation is however not only dependent on the physical model of the problem, but also on the next points:
- The size of the area in which the calculation takes place.
- The size and distribution of the elements
- The boundary conditions
The choices for these three points have to be evaluated with the problem that has to be solved in mind. These calculations are about the values and distribution of the water under-pressures in the shear zone and on the blade. A variation of the values for point 1 and 2 may therefore not influence this part of the solution. This is achieved by on the one hand increasing the area in which the calculations take place in steps and on the other hand by decreasing the element size until the variation in the solution was less than 1% (see figures 5 and 6).
The distribution of the elements
is chosen such that a finer mesh is present around the blade tip, the shear
zone and on the blade, also because of the blade tip problem.
A number of boundary conditions follow from the physical model of the cutting process, these are:
The boundary condition in the shear zone. This is described by equation (15).
The boundary condition along the free sand surface. The hydrostatic pressure, at which the process takes place, can be chosen, when neglecting the dimensions of the blade and the layer in relation to the hydrostatic pressure head. Because these calculations are meant to obtain the difference between the water vacuum pressures and the hydrostatic pressure it is valid to take a zero pressure as the boundary condition.
The boundary conditions along the boundaries of the area where the calculation takes place that are located in the sand package are not determined by the physical process. For this boundary condition there is a choice between:
- A hydrostatic pressure along the boundary.
- A boundary as an impermeable wall.
- A combination of a known pressure and a known specific flow rate.
None of these choices complies
with the real process. Water from outside the calculation area will flow through
the boundary. This also implies, however, that the pressure along this boundary
is not hydrostatic. If, however, the boundary is chosen with enough distance
from the real cutting process the boundary condition may not have an influence
on the solution. The impermeable wall is chosen although this choice is
arbitrary. Figure 3 gives an impression of the size of the area and the
boundary conditions, while figures 5 and 6 show the element mesh. Figures 2, 7
and 8 show the two-dimensional distribution of the water vacuum pressures,
while figure 9 shows the vacuum pressure distribution on the blade and in the
shear zone.
The blade tip
problem
During the physical modeling of the cutting process it has always been assumed that the blade tip is sharp. In other words, that in the numerical calculation, from the blade tip, a hydrostatic pressure can be introduced as the boundary condition along the free sand surface behind the blade. In practice this is never valid, because of the following reasons:
- The blade tip always has a certain rounding, so that the blade tip can never be considered really sharp.
- Trough wear of the blade a flat section develops behind the blade tip, which runs against the sand surface (clearance angle zero)
- If there is also dilatancy in the sand underneath the blade tip it is possible that the sand runs against the flank after the blade has passed.
- There will be a certain vacuum pressure behind the blade as a result of the blade speed and the cutting process.
A combination of these factors determines the distribution of the water vacuum pressures, especially around the blade tip. The first three factors can be accounted for in the numerical calculation as an extra boundary condition behind the blade tip. Along the free sand surface behind the blade tip an impenetrable line element is put in, in the calculation. The length of this line element is varied with 0.0hi , 0.1hi and 0.2hi. It showed from these calculations that especially the water vacuum pressures on the blade are strongly determined by the choice of this boundary condition as indicated in figure 10. Table 1 shows the dimensionless pore vacuum pressures.
Figure
10: The average
pore vacuum pressure on the blade and in the shear zone as a function of the
length of the flat wear zone w.
It is hard to estimate to what
degree the influence of the vacuum pressure behind the blade on the water
vacuum pressures around the blade tip can be taken into account with this extra
boundary condition. Since there is no clear formulation for the vacuum pressure
behind the blade available, it will be assumed that the extra boundary
condition at the blade tip describes this influence. The laboratory research of
Miedema (1987, [23]) has made this more evident.
Table 1: The dimensionless pore pressures p1m in the shear zone (s) and p2m on the blade surface (b) as a function of the blade angle a, de shear angle b, the ratio between the blade height hb and the layer thickness hi and the ratio between the permeability of the situ sand ki and the permeability of the sand cut kmax, with a wear zone behind the edge of the blade of 0.2hi.
|
|
hb/hi |
ki/kmax=1 |
ki/kmax=0.25 |
||||
|
|
b =30 |
37.5 |
45 |
30 |
37.5 |
45 |
|
|
a =15 |
1 (s) |
0.156 |
0.168 |
0.177 |
0.235 |
0.262 |
0.286 |
|
2 (s) |
0.157 |
0.168 |
0.177 |
0.236 |
0.262 |
0.286 |
|
|
3 (s) |
0.158 |
0.168 |
0.177 |
0.237 |
0.262 |
0.286 |
|
|
1 (b) |
0.031 |
0.033 |
0.035 |
0.054 |
0.059 |
0.063 |
|
|
2 (b) |
0.016 |
0.017 |
0.018 |
0.028 |
0.030 |
0.032 |
|
|
3 (b) |
0.011 |
0.011 |
0.012 |
0.019 |
0.020 |
0.021 |
|
|
|
b =25 |
30 |
35 |
25 |
30 |
35 |
|
|
a =30 |
1 (s) |
0.178 |
0.186 |
0.193 |
0.274 |
0.291 |
0.308 |
|
2 (s) |
0.179 |
0.187 |
0.193 |
0.276 |
0.294 |
0.310 |
|
|
3 (s) |
0.179 |
0.187 |
0.193 |
0.277 |
0.294 |
0.310 |
|
|
1 (b) |
0.073 |
0.076 |
0.078 |
0.126 |
0.133 |
0.139 |
|
|
2 (b) |
0.049 |
0.049 |
0.049 |
0.084 |
0.085 |
0.086 |
|
|
3 (b) |
0.034 |
0.034 |
0.033 |
0.059 |
0.059 |
0.059 |
|
|
|
b =20 |
25 |
30 |
20 |
25 |
30 |
|
|
a =45 |
1 (s) |
0.185 |
0.193 |
0.200 |
0.289 |
0.306 |
0.325 |
|
2 (s) |
0.190 |
0.198 |
0.204 |
0.304 |
0.322 |
0.339 |
|
|
3 (s) |
0.192 |
0.200 |
0.205 |
0.308 |
0.325 |
0.340 |
|
|
1 (b) |
0.091 |
0.097 |
0.104 |
0.161 |
0.174 |
0.187 |
|
|
2 (b) |
0.081 |
0.082 |
0.083 |
0.146 |
0.148 |
0.151 |
|
|
3 (b) |
0.067 |
0.065 |
0.063 |
0.120 |
0.116 |
0.114 |
|
|
|
b =15 |
20 |
25 |
15 |
20 |
25 |
|
|
a =60 |
1 (s) |
0.182 |
0.192 |
0.200 |
0.278 |
0.303 |
0.324 |
|
2 (s) |
0.195 |
0.204 |
0.211 |
0.314 |
0.339 |
0.359 |
|
|
3 (s) |
0.199 |
0.208 |
0.214 |
0.327 |
0.350 |
0.368 |
|
|
1 (b) |
0.091 |
0.103 |
0.112 |
0.158 |
0.184 |
0.205 |
|
|
2 (b) |
0.100 |
0.106 |
0.109 |
0.182 |
0.196 |
0.204 |
|
|
3 (b) |
0.094 |
0.095 |
0.093 |
0.174 |
0.176 |
0.174 |
|
ANALYTICAL WATER
PORE PRESSURE CALCULATIONS
As is shown in figure 4, the water can flow from 4 directions to the shear zone where the dilatancy takes place. Two of those directions go through the sand which has not yet been deformed and thus have a permeability of ki , while the other two directions go through the deformed sand and thus have a permeability of kmax. Figure 2 shows that the flow lines in 3 of the 4 directions have a more or less circular shape, while the flow lines above the blade have the character of a straight line. If a point on the shear zone is considered, then the water will flow to that point along the 4 flow lines as mentioned above. Along each flow line, the water will encounter a certain resistance. One can reason that this resistance is proportional to the length of the flow line and reversibly proportional to the permeability of the sand, the flow line passes. Figure 11 shows a point on the shear zone and it shows the 4 flow lines. The length of the flow lines can be determined with the equations 20, 21, 22 and 23. The variable "Lmax" in these equations is the length of the shear zone, which is equal to hi/sin(b), while the variable "L" starts at the free surface with a value zero and ends at the blade tip with a value "Lmax".
Figure 11: The flow lines used in the analytical method.
|
|
(20) |
|
|
(21) |
|
|
(22) |
|
|
(23) |
The total resistance on the flow lines can be determined by dividing the length of a flow line by the permeability of the flow line. The equations 24, 25, 26 and 27 give the resistance of each flow line.
|
|
(24) |
|
|
(25) |
|
|
(26) |
|
|
(27) |
Since the 4 flow lines can be considered as 4 parallel resistors, the total resulting resistance can be determined according to the rules for parallel resistors. Equation 28 shows this rule.
|
|
(28) |
The resistance Rt in fact replaces the hi/kmax part of the equations 13, 14, 18 and 19, resulting in equation 29 for the determination of the pore vacuum pressure of the point on the shear zone.
|
|
(29) |
The average pore vacuum pressure on the shear zone can be determined by summation or integration of the pore vacuum pressure of each point on the shear zone. Equation 30 gives the average pore vacuum pressure by summation.
|
|
(30) |
The determination of the average pore vacuum pressure on the blade cannot be carried out by integration or summation, because the calculation only gives the pore vacuum pressure at the tip (edge) of the blade. It is known that the pore vacuum pressure at the top of the blade equals zero, because the sand at that point is in direct contact with the surrounding water. If the pore vacuum pressure distribution on the blade is considered linear, then the average pore vacuum pressure equals 50% of the pore vacuum pressure at the blade edge.
|
|
(31) |
However figure 9 shows (left graph) that this distribution is not linear. Going from the tip (edge) of the blade to the top of the blade, first the pore vacuum pressure increases until it reaches a maximum and then it decreases (non-linear) until it reaches zero at the top of the blade. In this graph, the top of the blade is left and the tip of the blade is right. The graph on the right side of figure 9 shows the pore vacuum pressure on the shear zone. In this graph, the tip of the blade is on the left side, while the right side is the point where the shear zone reaches the free water surface. Thus the pore vacuum pressure equals zero at the free water surface (most right point of the graph).
Because the distribution of the pore vacuum pressure is non-linear, a shape factor has to be used. From the FEM calculations of Miedema [23] and Zhao [30] it is known, that the shape of the pore vacuum pressure distribution on the blade depends strongly on the ratio of the length of the shear zone and the length of the blade, and on the length of the flat wear zone (as shown in figure 10). A high ratio should result in a shape factor higher then 2, while a low ratio should result in a factor smaller then 0.5. Equation 32 gives the ratio in a modified form. The value of the power has been determined by trial and error.
|
|
(32) |
Appendix A shows the source code of a Visual Basic subroutine, calculating the dimensionless pore vacuum pressures similar to table 1 (from the FEM calculations). The subroutine also calculates the cutting forces according to Miedema [23]. The difference between table 1 and this analytical method is less then 10%.
Table 2: A comparison between the numerical and analytical calculated dimensionless pore vacuum pressures.
|
ki/kmax=0.25 |
p1m (table 1) |
p2m (table 1) |
p1m (analytical) |
p2m (analytical) |
|
a=30, b=30, hb/hi=2 |
0.294 |
0.085 |
0.333 |
0.072 |
|
a=45, b=25, hb/hi=2 |
0.322 |
0.148 |
0.339 |
0.140 |
|
a=60, b=20, hb/hi=2 |
0.339 |
0.196 |
0.338 |
0.196 |
CONCLUSIONS AND RECOMMENDATIONS
In the past decades many research has been carried out into the different cutting processes. The more fundamental the research, the less the theories can be applied in practice. The analytical method as described in this paper, gives a method to use the basics of the sand cutting theory in a very practical and pragmatic way. One has to consider that usually the accuracy of the output of a complex calculation is determined by the accuracy of the input of the calculation, in this case the soil mechanical parameters. Usually the accuracy of these parameters is not very accurate and in many cases not available at all. The accuracy of less then 10% of the analytical method described in this paper is small with regard to the accuracy of the input. This does not mean however that the accuracy is not important, but this method can be applied for a quick first estimate.
By introducing some shape factors to the shape of the streamlines, the accuracy of the analytical model can be improved. Figure 4 shows the more or less circular shape of the streamlines. But more or less is not exactly, so a shape factor would improve the accuracy.
REFERENCES
LIST OF SYMBOLS USED
|
A |
surface |
m |
|
b |
width of the blade of blade element |
m |
|
e |
volume strain |
% |
|
Fci |
cutting force (general) |
kN |
|
g |
gravitation acceleration |
m/s |
|
hi |
initial layer thickness |
m |
|
k |
permeability |
m/s |
|
ki |
initial permeability |
m/s |
|
kmax |
maximum permeability |
m/s |
|
km |
effective permeability |
m/s |
|
l |
length of the shear zone |
m |
|
n |
normal on an edge |
m |
|
ni |
initial pore
percentage |
% |
|
nmax |
maximum pore
percentage |
% |
|
p |
pressure (pore
pressure) |
kPa |
|
patm |
atmospheric pressure |
kPa |
|
pcalc |
calculated dimensionless pressure (pore pressure) |
- |
|
pdamp |
saturated vapor pressure (12 cm water column) |
kPa |
|
preal |
real acting pressure (pore pressure) |
kPa |
|
p1m |
average pore pressure in the shear zone |
- |
|
p2m |
average pore pressure on the blade |
- |
|
q, q1
,q2 |
specific flow rate |
m/s |
|
Q |
flow rate per unit blade width |
m/s |
|
s |
length of a flow line |
m |
|
s |
measure for the layer thickness |
m |
|
t |
time |
s |
|
Dt |
time interval |
s |
|
vc |
cutting velocity perpendicular on the blade edge |
m/s |
|
V |
volume increase per unit of blade width |
m |
|
x |
coordinate |
m |
|
y |
coordinate |
m |
|
z |
coordinate |
m |
|
z |
water depth |
m |
|
a |
blade angle |
rad |
|
b |
shear angle |
rad |
|
j |
angle of internal friction |
rad |
|
d |
soil/steel angle of friction |
rad |
|
rw |
water density |
ton/m |
APPENDIX A: The Visual Basic Subroutine
Sub Pressure(Fh, Fv, P1m, P2m, Factor)
Dim N As Integer, I As Integer
Dim Lmax As Single, L As Single, StepL As Single
Dim P As Single, DPMax As Single, DP As Single, DP0 As Single, P0 As Single
Dim Flag As Boolean, Argument As Single
Dim S1 As Single, S2 As Single, S3 As Single, S4 As Single
Dim R1 As Single, R2 As Single, R3 As Single, R4 As Single, Rt As Single
Dim W1 As Single, W2 As Single, K2 As Single
Dim CoefNC As Single, CoefC As Single
Dim C1 As Single, C2 As Single, D1 As Single, D2 As Single
Dim Teta1 As Single, Teta2 As Single, Teta3 As Single, Teta4 As Single
Teta1 = Pi / 2 - Alpha - Beta
Teta2 = Alpha + Beta
Teta3 = Pi - Beta
Teta4 = Pi + Beta
N = 100
Lmax = Hi / Sin(Beta)
StepL = Lmax / N
P = 0
DPMax = RhoW * G * (Z + 10)
Flag = False
For I = 0 To N
L = I * StepL + 0.0000000001
S1 = (Lmax - L) * Cos(Teta1) * Pi / 2 + (Lmax - L) * Sin(Teta1) + Hb / Sin(Alpha)
S2 = L * Teta2
S3 = L * Teta3
S4 = (Lmax - L) * Teta4 + 0.1 * Hi * Pi
R1 = S1 / Kmax
R2 = S2 / Kmax
R3 = S3 / Ki
R4 = S4 / Ki
Rt = 1 / (1 / R1 + 1 / R2 + 1 / R3 + 1 / R4)
DP = RhoW * G * Vc * E * Sin(Beta) * Rt
If I = N Then DP0 = DP
P0 = P0 + DP
If DP > DPMax Then
DP = DPMax
Flag = True
End If
P = P + DP
Next I
P1m = (P - DP / 2) / N
P0 = (P0 - DP0 / 2) / N
Factor = (Hi / Hb) ^ (Pi / 2 - Alpha * 1.2) * Sin(Alpha + Beta) * Sin(Alpha) / Sin(Beta) / 2
If Flag Then
Argument = -2 * Factor * (P0 - P1m) / P1m
Factor = Factor * Exp(Argument) + (1 - Exp(Argument))
End If
P2m = DP * Factor
If P2m > DPMax Then P2m = DPMax
W1 = P1m * Hi * B / Sin(Beta)
W2 = P2m * Hb * B / Sin(Alpha)
K2 = (W1 * Sin(Phi) + W2 * Sin(Alpha + Beta + Phi)) / Sin(Alpha + Beta + Phi + Delta)
Fh = K2 * Sin(Alpha + Delta) - W2 * Sin(Alpha)
Fv = K2 * Cos(Alpha + Delta) - W2 * Cos(Alpha)
P1m = P1m * Kmax / (RhoW * G * Vc * E * Hi)
P2m = P2m * Kmax / (RhoW * G * Vc * E * Hi)
CoefC = RhoW * G * (Z + 10) * Hi * B
D1 = Fh / CoefC
D2 = Fv / CoefC
CoefNC = (RhoW * G * Vc * E * Hi ^ 2 * B) / ((Ki + Kmax) / 2)
C1 = Fh / CoefNC
C2 = Fv / CoefNC
End Sub