Dr.ir. S.A. Miedema

Abstract.
Introduction.
The theoretical cutting force model.
The forces on the dredging wheel.
The direction of the cutting force.
The influence of the shear angle.
Application of the theory.
Conclusions.
Bibliography.
List of Symbols used.

During the last decades extensive research has been carried out into the fundamental process of dredging. In the Laboratory "The Technology of Soil Movement" of the Delft University of Technology research has been carried out into the forces generated when cutting densely compacted sand under water.

In 1974 Joanknecht [4] discerned the importance of the pore pressures and the average dilatation "e" in 1977, but he did not give a method to determine the pore pressures.

The determination of the values of these pore pressures only became possible when the necessary software (finite element method) became available.

Based on the pore pressure calculations a mathematical model of the cutting forces was set up by Miedema [12].

Equations giving the forces and power requirements for the use of a dredging wheel will be derived from the equations upon which the cutting force model is based. It is possible to expand these equations and to make them suitable for a numerical solution, but those given in this publication will suffice for a preliminary approach and a better insight in the cutting process. The cutting theory was verified by research into the cutting forces on straight blades.

For many years the Laboratory the Technology of Soil Movement has been engaged in fundamental and applied research into the nature of various dredging processes.

In this research attention has been paid to the cutting processes in sand, clay and rock.

A useful result of this research was the development of the disc bottom cutterhead by Bos and Joanknecht [5] .

Towards the end of the seventies, as part of the research of the Offshore Technology Group of the Delft University of Technology, research into the behaviour of sea going cutter section dredgers was started. This led to the development of the computer programme DREDMO in 1983 (see Keuning and Journee [6] , de Koning, Miedema and Zwartbol [7] and Miedema [9,11] ) . The behaviour of sea going cutter suction dredgers is largely determined by the soil/cutterhead interactions, so research into this aspect was necessary.

Unfortunately, at the time when this research started there was still no adequate mathematical model available for the determination of the cutting forces in densely compacted sand under water. In the seventies however, it was known that the cutting forces in saturated sand were determined by under pressures in the pore water resulting from dilatation (Joanknecht [4] and van Os [14] ) .

Based on this knowledge a theoretical model was developed for cutting with straight blades in densely compacted sand under water. With the aid of characteristic values for the process determining parameters a model was developed for the forces and the torque of a cutter head. The implementation of this model in DREDMO called for a number of simplifications owing to the nature of the computer programme (time domain simulations). Details of the simplified model of the cutting forces were published by Miedema in 1984 [10], while the theoretical background was explained in 1985 [12].

In the near future extrapolation of the cutting theory will make it possible to determine the behaviour of a sea going dredging wheel dredger with the DREDMO programme.

In this paper an attempt is made to apply the cutting theory to a dredging wheel.

It is assumed that sharp blades with a positive free running angle are used and that there is no cavitation in the pore water during the cutting process.

The cutting process in densely compacted sand under water is determined by the dilatational properties of the sand. Additional factors influencing the process are:

• the permeability of the sand
• the angle of internal friction of the sand
• the soil/interface friction angle
• the geometry of the cutting process

Figure 1 gives a schematization of the cutting process. The most important parameters are shown in this figure. During the deformation of the sand in the shear zone, the pore volume increases (dilatation). Water must flow in from the surface of the sand to fill this increased volume. This flow of water causes under pressure in the pore water, as a result of which the grain stresses increase and with this the forces required to cut the sand. At high cutting velocities and thus high velocities of the water flowing through the pores, the under pressure becomes so great that saturated water vapour pressure is reached. Cavitation then occurs in the deformation zone. Increasing the cutting velocity will expand the area where the cavitation occurs, with a limit where cavitation occurs on the entire blade. The degree to which under pressure occurs in the pore water depends on the cutting velocity, the thickness of the cut layer and the geometry of the cutting process, but also on soil mechanical parameters (above all the permeability).

Types of sand producing quick and slow cavitation can be distinguished although no distinct boundary can be given between them. In addition the occurrence of cavitation also depends upon the depth at which dredging is being performed. This process has already been described by van Os [14], Miedema [10,12], Steeghs [15] and Brakel [1], while the influence of the soil mechanical parameters has been described by van Leussen and Nieuwenhuis [8]. From this it can be deduced that the cutting forces are proportional to the cutting velocity, the increase in the pore volume, the square of the cut layer thickness, the length of the cutting edge and inversely proportional to the permeability of the sand. If it is assumed that no cavitation occurs in the sand, the following general cutting formula (Miedema [12]) can be derived:

(1)

The proportionality coefficient c_i includes the influences of:

• the under pressure on the blade
• the under pressure in the deformation zone
• the blade angle
• the angle of internal friction of the sand
• the soil/interface friction angle
• the ratio h_b/h_i
• the rate of wear of the blade.

For the horizontal component of the force the value of c₁ can be found in figure 2 and for the vertical component c₂ can be found in figure 3. These figures are valid for cases in which the blade height h_b is equal to the thickness of the layer cut h_i and no cavitation occurs in the sand. A weighed average between the permeability of the undisturbed sand and the permeability of the deformed sand above the blade should be chosen for the permeability k_m of the sand. A reasonable estimate can be made by taking the average of the two permeability coefficients. A more accurate determination of k_m can be obtained with the equation below:

With: a1 + a2 = 1

(2)

Table 1 gives some of the values for a₁ and a₂. From this table it appears that a₁ and a₂ depend upon the geometry of the cutting process.

Table 1

The coefficients are still, though to a lesser degree, dependent upon the soil mechanical properties of the sand. The value of a₁ varies between 0.4 and 0.6.

For a straight blade the parameters resulting from the geometry of the cutting process are more or less fixed and it is thus useful to be able to make a more accurate estimate of k_m.

The complex geometrical cutting processes of the cutter head and the dredging wheel cause variations in the thickness of the cut layer, the cutting velocity and the blade angle in relation to both time and space. An accurate estimate of k_m is then only possible using a numerical method to determine the cutting forces.

For the method using the characteristic parameters, the characteristic value of k_m must be determined, which goes with the characteristic cut layer thickness, blade angle etc.

Fig. 1: The plane strain cutting process.

Fif. 2: The coefficient c₁ as a function of d and f.

Fig. 3: The coefficient c₂ as a function of d and f.

In order to render formula 1 useful for the calculation of cutting forces and power requirements for the dredging wheel it is first necessary to analyze the geometry of the cutting process in the dredging wheel. Figures 4 and 5 give a picture of the cutting process in the axial plane and the median plane of the dredging wheel.

The cutting velocity, the cut layer thickness and the length of the cutting edge are a function of the haulage speed, the number of revolutions, the length of the step, the dimensions of the dredging wheel and the height of the slope, whilst the length of the cutting edge also varies with time. Unlike the cutter head the dredging wheel has a cut layer thickness which is constant in time, but which is dependent upon its position on the cutting edge (fig. 4). The following equation can be derived for the thickness of the cut layer as a function of the distance r to the axis of the dredging wheel:

(3)

(4)

From figure 5 it can be seen that when the blade enters the slope, only the outermost part of the cutting edge is involved in the cutting process. The length of the cutting edge which is involved in the cutting process is thus not constant in time. The cutting velocity is also at its greatest on the outer circumference on the dredging wheel, so that the greatest cutting forces are also developed there (eqn. 1). This partially explains for the fact that the greatest wear occurs on the outer circumference.

For the cutting force model it is important to find a characteristic cutting velocity, cut layer thickness, length of the cutting edge and radius for the dredging wheel.

Because there is a linear increase in cutting forces proportional to the increase in cutting velocity, the following can be derived for the average cutting force:

With: W = w t

(5)

Provided that the dynamical blade angle and the cut layer thickness are constant over the cutting edge which is approximately the case.

For the area of the longitudinal cross section:

(6)

can be derived. The radius r(W ) is the distance from the axis of the dredging wheel to the breach (fig. 6) . From the equations 5 and 6 it follows:

(7)

Now a radius R₁ can be defined so that the area of the part of the circle segment in figure 6 equals the area of the longitudinal cross section. With:

(8)

and

(9)

It follows that

(10)

This gives for the characteristic radius:

(11)

Hence:

(12)

so that for the characteristic cutting velocity:

(13)

is valid.

The characteristic length of the cutting edge is now:

(14)

The characteristic cut layer thickness follows from equation 2 when the characteristic radius is substituted for the radius r (eqn. 3). Substituting the characteristic values for the cut layer thickness, the blade width and the cutting velocity in equation 1 gives the average force on a blade as long as this is in contact with the breach. To determine the total resultant force the following is valid:

(15)

Fig. 4: The cutting process in the axial plane of the dredging wheel.

Fig. 5: The cutting process in the median plane of the dredging wheel.

Fig. 6: The substituting circle segment.

The cutting force on the dredging wheel can be resolved into a tangential component and an axial component (fig. 7). The tangential component is more or less equivalent to the horizontal force on a straight blade, while the axial component is equivalent to the vertical force on the straight blade (fig. 1).

A correction for the direction of the cutting velocity is necessary if the angle resulting from the circumferential velocity and the haulage velocity becomes too great (fig. 7). The tangential and axial forces can be determined by substituting the characteristic cut layer thickness, length of the edge involved in the cutting process and cutting velocity in equation 1 and by determining the coefficient c_i for both forces with the aid of figures 2 and 3. The blade angle will in fact never be precisely 30 or 45 . An interpolation or extrapolation (a <30 ) is then necessary.

Linear interpolation or extrapolation is obtained by determining the values of c_i for a =30 and a =45 and drawing a straight line through them (fig. 8).

A greater degree of accuracy can be obtained by determining the values of c_i for a =30 , a =45 and a =60 and drawing a parabola through them (fig. 8).

The torque of the drive can be now determined by the use of:

(16)

so that for the driving power:

(17)

is valid. Theoretically for the haulage power

(18)

is valid.

It is noteworthy that the axial force has the same direction as the haulage velocity.

This indicates that the cutting process produces haulage power. In fact this is only valid in cases where there is a pure cutting process. With blade angles greater than 40

the direction of the axial force can reverse. Because the blade angle of the dredging wheel extends form 45 where the radius is the smallest, to 30 at the outer edge of the wheel, it may be possible that the axial force changes direction on the blade. This also depends on the soil mechanical parameters.

A numerical determination of the cutting forces can provide more information about this. As soon as there is any question of a run up of the sand to the back of the blade (negative free running angle) the direction of the axial force can reverse and power is required for the haulage.

By means of measurements in two types of soil in the field, van Drimmelen en Eijgenraam [2] have established that the part of the haulage force resulting from the cutting force may be positive or negative.

From the cutting theory it is not yet possible to determine the influence of wear (dullness of the cutting edge) but it is probable that the axial force will also be reversed in this case.

Fig. 7: The direction of the cutting forces.

Fig. 8: Interpolation to gain the coefficients c₁ and c₂ ( linear above, quadratic below).

In the dredging wheel there is a certain aperture between the buckets. The blade angle is greater than the angle which the blade makes with the vertical. A consequence of this is that, depending on the thickness of the cut layer and the shear angle, it is possible for the cut layer to run up against the back of the preceding bucket. This is shown in figure 9. In this case the thickness of the deformed cut layer is equal to the aperture d between the buckets. The cut layer thickness is determined by the haulage velocity and the number of revolutions of the dredging wheel and is thus a value which can be directly influenced.

The angle of shear depends not only on the geometry of the cutting process but also on the soil mechanical properties of the sand. In the theory of the cutting of dry soil use is commonly made of a shear angle of 45-¹/_2f with the horizontal. From modern soil mechanics it appears that a shear angle of 45-¹/_2n is also possible. Both shear angles can be observed experimentally (Vermeer [16]). In this it is assumed that the main stress directions are horizontal and vertical. This is not the case for the cutting process and thus it cannot be assumed that the shear angles mentioned above will be found. One way of determining the angle of shear is to select the value at which the change in the energy absorbed in the sand due the volume changes and transformed into heat is minimized (Rowe [17]). This will be described in a future publication. The shear angle is then a function of the blade angle, the angle of soil/interface friction, the angle of internal friction and the ratio between the height of the blade and the thickness of the cut layer. Figure 10 gives the shear angles for the case that h_b = h_i .

From this theory in fact it is possible to determine the angle of shear for all blade angles up to 60 . With a blade angle greater than 60 the mechanism of the cutting process changes. In dry soil this can be demonstrated by the "wedge theory" of Hettiaratchy and Reece [3].

It is possible to show the "wedge theory' theoretically for the cutting of densely compacted sand under water but further consideration of this is beyond the scope of this paper.

Fig. 9: A run up against the back of the preceeding bucket.

To illustrate the theory which has been described an example of a calculation of the forces on a dredging wheel will be given. The information relating to the dredging wheel is taken from the publication of Eijgenraam and van Drimmelen [2].

Soil mechanical properties of the sand.

Dredging wheel.

This gives:

From this follows:

From the initial cut layer thickness, blade angle and shear angle, the thickness of the deformed cut layer is:

(19)

This gives:

h_def = 0.159 m

Here it is still possible to speak of a pure cutting process. Increasing the haulage velocity can cause a run up against the back of the preceding bucket, depending on the aperture. With the aid of figure 10 it is possible to calculate the thickness h_def and thus at which haulage velocity the sand will run up against the back of the preceding bucket. It must be realized that other types of soil will produce other demands with regard to the means of cutting. However, the extrapolation of the cutting theory for straight blades to the dredging wheel increases the insight into the complicated cutting process which is involved. The choice must be made between a generalized model or a detailed numerical model is principally determined by the use to which the model will be put. In the design phase of a dredging wheel it is useful to know with the greatest possible degree of accuracy not only the magnitude of the forces on the edge of the blade, but also their distribution. For this purpose a numerical model is most suitable. The detailed numerical method is too time consuming for computer programmes such as DREDMO, which are intended for the determination of the behaviour of dredging vessels (cutter suction dredgers and dredging wheel dredgers) in swell. In addition small variations in the total cutting forces have little effect upon the movement of the dredger. In such cases a generalized cutting force model (like the model presented in this paper) is more suitable. Both types of model are receiving attention within the framework of the research in the Laboratory the Technology of Soil Movement.

Fig. 10: The shear angle b as a function of d and f.

A close correlation was obtained between the forces, the pore pressures and the shear angles which were measured and those obtained by calculation ( Miedema [12] ).

With the equations derived it is possible to make an estimate of the forces and power requirements, caused by the cutting process on a bucket wheel.

It is also possible to determine the thickening of the layer cut on the blade.

The latter is especially important in relation to the possibility of the aperture between the buckets becoming clogged.

It is difficult to determine the haulage force accurately because this force can change direction not only as a result of wear but also in consequence of variations of the soil mechanical parameters.

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Last modified Friday June 09, 2000 by: Sape A. Miedema

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

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