Dr.ir. S.A. Miedema

Abstract.
Pore Water, Shear Stress and Cutting.
Application of the Cutting Force Model to a Cutterhead.
Selection of the Characteristic Parameters.
Verification of the Cutting Force Model.
Conclusions.
Bibliography.
List of Symbols used.

The use of dredgers to work at sea is strongly dependent upon the ability to work in waves of various heights and lengths. As soon as the critical wave height is reached for a particular vessel, work ceases, so there is general interest in an improved performance with a consequent reduction in down time. In order to extend the possibility of using floating dredgers as opposed to semi-submersibles or fixed platforms, the application of swell compensators for the excavating element, for example, cutter heads, is being attempted. This is not so simple as it might appear. In order to design swell compensators for the excavating element of dredgers, it is essential to have some understanding of the interactions such as those between cutter head and soil when cutter suction dredgers are working at sea. Both qualitative and quantitative predictions relating to this interation must be made. This article describes an empirical, physical cutting model of the cutting forces and gives some results of model tests with a cutter head in densely compacted sand.

To optimize the dredging process it is important to understand the cutting process. This will enable predictions to be made in relation to the cutting forces which will be encountered when different types of excavation elements are being used.

The literature on this subject indicates that much is already known about the cutting of dry sand. The models of Reece^{1, 9} and Osman⁸ are well known, but is doubtful whether these models are also valid for densely packed sand which is cut under water. When investigating this, it was assumed that the rate at which the sand is deformed during the cutting process, was many times greater than the permeability of the sand, (the k value).

In addition, it was assumed that the deformation rate was so small that the inertial forces could be neglected.

The influence of the deformation rate upon the cutting forces is caused by the presence of a varying amount of pore water, the volume of which is related to the changes in the pore volume in consequence of changes in the shear stress in the sand.

The phenomenon of dilatancy is especially important in this respect. In sand which is not completely saturated with water, the amount of air in the pores can easily follow the volume changes because of the compressibility of the air (law of Pascal). This hardly influences the pore pressures. If cutting takes place under water, the inflowing water (which is compressible) will encounter some resistance, resulting in the creation of a decrease in pressure in the pore water in the shear zone, compared to the hydrostatic pressure in the location of the shear zone.

For a two-dimensional cutting process (blade length>>thickness of the layer cut) the following can now be derived:

Figure 1: The two-dimensional cutting process.

If a blade moves a distance of Dx in the sand (fig. 1), then for the sand element in the shear zone:

(1)

In which:

The volume of the water that must flow into the sand element now equals:

(2)

In which:

This is equivalent to:

(3)

Hence:

(4)

From the theory of soil mechanics¹⁰ and the groundwater mechanic¹¹, it is known that:

(law of Darcy)

(5)

In which:

With:

(6)

In which:

Thus:

(7)

And:

(8)

Substituting this in equation (4) gives:

(9)

The average pressure drop in the shear zone can be found by integrating equation (9) over the shear zone, this gives:

(10)

In which:

The fall in pressure is governed by a limiting factor. For a specific product of cutting velocity and cut layer thickness, the drop in pressure will be so great that the absolute pressure in the pores of the sand element may reach saturated water vapour pressure, dependent upon the prevailing environmental conditions. With an environmental temperature of 10 C, this water vapour pressure is about 12cm water column and can thus be disregarded in comparison to the atmospheric pressure. If the saturated water vapour pressure in the pores is reached, cavitation will occur. The pressure in the pores cannot decrease further and the pressure drop Dp remains constant with an increasing cutting velocity (in the sand element in question).

Figure 2: Dilatancy during the cutting process.

From figure 2 it can be seen that the distance between a sand element in the shear zone and the free sand surface depends upon the location of the sand element in the dilatancy zone. This means that cavitation begins where the effective path of the water is the longest. This will be in an area just in front of the edge of the blade (not at the edge of the blade). With an increase in cutting velocity, the cavitation zone will extend until it includes almost the entire dilatancy zone.

In the development of cavitation in the shear zone, three phases can now be distinguished.

Phase 1: The cutting velocity is low and/or the cut layer is thin. Pressure decreases in the pores, but there is still no cavitation. There is a linear decrease in the average pressure with increasing cutting velocity and the increasing layer thickness.

(11)

Phase 2: A transitional stage. The cutting velocity increases further and/or the cut layer becomes thicker. Local cavitation develops in the dilatancy zone. With a further increase in cutting velocity and/or in layer thickness, the cavitation zone continues to extend. The pressure no longer decreases in a linear relation to the cutting velocity and the cut layer thickness, but in accordance with the following equation.

(12)

Phase 3: Cavitation occurring throughout almost the entire dilatancy zone. With increased cutting velocity and/or cut layer thickness, the decrease in pressure remains more or less constant. At very high cutting velocities, cavitation can even occur behind the edge of the cutting blade.

(13)

In which:

Figure 3: Under pressure in the pores as a function of cutting velocity, cut layer thickness and the water depth.

These three phases are shown in figure 3 and can be approximated by the following equation:

(14)

This approximation is also evident from figure 3.

The exponents a1 and a2 can vary between 0 and 1. When the cohesion and the initial grain stress are neglected, the average shear stress in the dilatancy zone can be demonstrated by:

Thus:

(15)

In which:

For the cutting force in phase 1, the following is valid:

(16)

If localized cavitation occurs (phase 2) the following is valid:

(17)

With fully developed cavitation throughout the dilatancy zone (phase 3), the following equation is valid:

(18)

The three phases can be approximated by the following equation:

(19)

In which:

This equation forms the basis for the method of calculating the cutting forces required for the excavation of densely compacted sand with a cutter head.

When using this model to make an estimate of the cutting forces, which are obtained in practice, it is necessary to know the cutting velocity, the cutting direction, the cut layer thickness, the length of the blade involved in cutting and the permeability coefficient of the sand upon which the blades are acting.

As the cut layer thickness and the cutting velocity are a function, amongst others, of the position of the blade in the breach, the shape of the cutter head and of time, and as the permeability coefficient is not necessarily constant, it is not easy to make an exact calculation of the forces acting on the cutter head. For example, the permeability after shear is greater than the permeability before shear. But it is possible by a good choice of characteristic layer thickness, length of the blade involved in cutting and the cutting velocity, to obtain a good approximation of the loads on the cutter head by substituting the characteristic values in equation (19). The relation which is found, however, is not purely theoretical.

Photo 1: A view of the laboratory test stand.

In selecting the characteristic cut layer thickness, cutting velocity and length of the blade involved in cutting, it is important that these three parameters can be easily derived from a number of known values. The cutting velocity is determined by the number of revolutions of the cutter head, the swing speed of the cutter suction dredger and the diameter of the cutter head. In fact, the velocity at any one time is the sum of the vectors of the momentary circumferential velocity of a blade and the swing velocity. The circumferential velocity changes direction continually so the velocity of a blade is not constant in time, with respect to direction or speed (fig. 4). As the swing velocity is 6 to 10 times smaller than the circumferential velocity, it will have little influence upon the cutting velocity. The circumferential velocity of the cutter head is selected for the characteristic cutting velocity.

(20)

In which:

Figure 4: The variation of the cutting velocity.

The cut layer thickness is also determined by the number of revolutions, the cutter diameter, the swing velocity and the number of blades on the cutter head. Figure 5 illustrates how the cut layer thickness varies as a function of time and the position of the blade. This figure also shows the epicyclic path of the blades for undercutting and overcutting. The figure illustrates how the cut layer thickness decreases for the overcut, from a maximum value, when the blade enters a breach, to almost 0 at the moment that a blade leaves the breach. For undercutting, the reverse is true. When undercutting, the shape of the cut layer is clearly different from the shape formed when overcutting (fig. 6), which is of consequence with regard to the cutting forces.

The area of the cross section cut, however, is characteristic for each blade. This is shown in figure 7. This area can be determined by:

(21)

In which:

Figure 5: The epicyclic path of the blades.

The path followed by the blade in the breach, if the effect of the swing speed is discounted, is:

(22)

Thus the average cut layer thickness is:

(23)

In which:

Since, with a particular ratio between swing velocity and number of revolutions, the shape of the cut layer is fixed (fig. 6), equation (23) can be used for a characteristic cut layer thickness, figure 6 also shows that as the ratio between the circumferential velocity and the swing velocity increases, the shapes of the cut layers formed during overcutting gradually become more similar.

The part of the blade involved in cutting depends on size and shape of the blades and on the cross section of the cutter head in the breach (the theoretical cut surface as in fig. 8). Apparently, the theoretical cut surface in the working area increases almost linearly with the height of the breach and the size of the step. A reasonable assumption would be that there is a linear increase in the cutting blade length in relation to the theoretical cut surface. For the characteristic cutting length of the blade, the following equation is given:

(24)

In which:

For a disc-bottom cutter head with vertical blades it can be shown that the actual cutting blade length is the same as the characteristic cutting blade length. For all other cutter heads the actual cutting blade length is greater.

With the calculated characteristic cutting velocity (eqn. 20), cut layer thickness (eqn. 23) and cutting blade length (eqn. 24) substituted in equation 19, a provisional cutting force model can be set up. The only unknown parameter in this model is the permeability coefficient. Other soil mechanical parameters are included in the proportionality constants, such as angle of internal friction and dilatancy angle. If it is assumed that localized cavitation occurs in the cut layers and across the length of the blade, then equation 25 is valid.

(25)

In which:

For the driving torque, equation 26 is valid:

(26)

The term (r g Dn) of equation 19 is included in the proportionality constants.

Figure 6: The shape of the cut layer.

To test the validity of the cutting force model a series of tests was carried out in the laboratory "The Technology of Soil Movement" of the Delft University of Technology, using compacted sand with a d50 of 150 mm and a cone resistance of around 7 MPa. From these tests the exponents of the characteristic cut layer thickness, cutting velocity and the coefficient of proportionality were determined for seven cutter heads (scale 1:6, diameter 400 mm).

Table I shows a summary of the overcutting tests and Table II of the undercutting tests. Photos 2 and 3 show two of the cutter heads which were used in the tests.

During the laboratory tests it also appeared that for the type of sand used the cutting forces and the driving torque increased almost linearly with the cone resistance of the sand. It was therefore decided to make provisional modifications to the model to give;

(27)

and

(28)

In which:

The tables I and II refer to these equations.

Figure 7: The area of the cross-section of a cut layer.

Figure 8: The theoretical cut surface.

Photo 2: A normal crown cutterhead used in the tests.

Photo 3: A flattened cutterhead with disc wheel shaped blades used in the tests.

There was a good correlation between the mathematically derived cutting force model and the results of the laboratory tests. Although a number of simplifications are applied, it appears that the model is reliable and that it can be generally applied to cutter heads, with the proviso that if an accurate prediction of the cutting forces is to be made, the coefficients and exponents for the equations must be determined by model tests for each type of cutter head.

These equations lie at the basis of extended equations in which the effect of axial and radial movements of the cutter head, perpendicular to the swing direction, have been included.

These extended equations are implemented in the computer program "DREDMO" with which the nonlinear behaviour of a cutter suction dredger in swell can be determined (3, 4, 5 and 6). "DREDMO" can be used to investigate the effectiveness of various types of swell compensators on cutter suction dredgers. The main issue, however, is the soil cutter head interaction on which the behaviour of each swell compensator depends. It is thus very important to have a good understanding of the cutting process and to be able to make an estimate of the cutting forces.

1. Hettiaratchi, D.R.P. & Reece, A.R., "Symmetrical Three-dimensional Soil Failure". J. Terramech. (1964)4, (3), pp45-67.
2. Joanknecht, L.W.F., "A Review of Dredge Cutter Head Modelling and Performance". Proc. WODCON VII, San Francisco, Califonia, USA., 1976.
3. Keuning, P.J. & Journee, J., "Calculation Method for the Behaviour of a Cutter Suction Dredger Oparating in Irregular Waves". Proc. WODCON X, Singapore 1983.
4. Koning, J.de & Miedema, S.A. & Zwartbol, A., "Soil/Cutterhead Interaction under Wave Conditions". Proc. WODCON X, Singapore, 1983.
5. Miedema, S.A., "De interactie tussen snijkop en grond in zeegang". Proc. Baggerdag 19/11/1982, T.H. Delft, 1982.
6. Miedema, S.A., "Computersimulatie baggerschepen". De Ingenieur, Dec.1983. (Kivi/Misset):
7. Os, A.G. van, "Behaviour of Soil when excavated under water". International Course Modern Dredging. June 1977, The Hague, The Netherlands.
8. Osman, M.S., "The Mechanics of Soil Cutting Blades". J.A.E.R. 9 (4). Pp.313-328, 1964.
9. Reece, A.R., "The Fundamental Equation of Earth Moving Machinery". Proc. Symp. On Earth Moving Machinery. Institute of Mechanical Engineering, London, 1965.
10. Terzaghi, K. & Peck, R.B., "Soil Mechanics in Engineering Practise". John Wiley & Sons, Inc., 1967.
11. Verruijt, A., "Theory of Groundwater Flow". Macmillan, London, 1970.

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Last modified Sunday April 30, 2000 by: Sape A. Miedema

Copyright April, 2000   Dr.ir. S.A. Miedema

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