Dr.ir. S.A. Miedema
S. Becker BSc
Ir. P.S. de Jong
Ir. S. Wittekoek

Abstract.
Introduction.
The History of Clamshell Research.
The Operation and Kinematics of a Clamshell.
The Equations of Motion of a Clamshell.
The Forces Exerted on the Buckets by Sand.
The Research Carried Out.
Conclusions.
Bibliography.
List of Symbols used.

The literature reveals little about the prediction of the closing process of clamshell dredging buckets when cutting sand or clay under water. The results of research carried out, mostly relates to the use of clamshells in dry bulk materials.
While good prediction of the forces (in dry materials) involved are possible by measuring the closing curve, the very prediction of the closing curve of clamshells in general, seems to be problematic.
Because the dredging business is concerned with water saturated sand or clay has to be dredged, the research into the closing process of clamshell grabs had to start from scratch (except for the kinematics of clamshells). In 1989 the research carried out by Great Lakes Dredge & Dock Company resulted in a numerical method of calculating the closing process of clamshell grabs in water saturated sand and clay, which simulates the closing of a clamshell so that production and forces can be predicted. The calculation method is based on the non-linear equations of motion of the buckets and the sand cutting theory Miedema presented at WODCON XII. A clay cutting theory is implemented in the numerical model but will not be taken into consideration in this paper. In 1991, Great Lakes and the Delft University of Technology carried out laboratory research in which a scale model clamshell was used.
This research, carried out in dry and in water saturated sand, resulted in a verification and validation of the calculation method with respect to the closing curve, the angular velocity and the pulling force in the closing wire.
This paper contains results of a literature survey, the equations of motion of a clamshell grab, background to the sand cutting theory, results of the computer program CLAMSHELL, and it will give some of the results of the research carried out with respect to verification and validation of the computer program.

It is important for dredging contractors to be able to predict the production of their dredges. Many studies have been carried out with respect to cutter suction dredges and hopper dredges. From the literature it became clear that, although many researchers have investigated the closing process of clamshell grabs, no one had succeeded in predicting their closing process.

Figure 1: The largest clamshell grab used in dredging, the Chicago, in full operation.

Since many clamshell grabs are being used in dredging industry in the U.S.A., it is important to have a good prediction of the production of clamshells in different types of soil. This was the reason for Great Lakes to start fundamental research into the processes involved in the digging of clamshell grabs in cooperation with dr.ir. S.A. Miedema. In 1989 this resulted in the computer program CLAMSHELL [9], which simulates the digging process of clamshell grabs in water saturated sand and clay. Although the results of the program were promising, there was a need for verification and validation of the program by means of measurements. Model research was carried out at the Laboratory of Soil Movement of the Delft University of Technology in 1991, Wittekoek [21]. The results of the measurements correlate very well with the computer program. The program is used by Great Lakes for production estimates and as well for the design of new clamshell grabs. Figure 1 shows the largest clamshell grab used in dredging, the Chicago, owned by Great Lakes Dredge & Dock Company. Figure 2 shows the 50 cubic yard
clamshell of the Chicago. Figure 3 shows the clamshell against human size.

Figure 2: The 50 cubic yard clamshell buckets.

The first grab reported was designed by Leonardo da Vinci (1452-1519) in the 15th century. Although the basic working principles remained the same, grab designs have improved dramatically as a result of trial and error, though research has had some influence. The following reviews some of the results found of research carried out in this century.

Figure 3: The clamshell buckets versus human size.

Pfahl 1912 [14] investigated the influence of the deadweight of a grab with respect to the payload for grabs of 1 m to 2.25 m. He concluded that the payload has a linear relation with the deadweight.
Ninnelt 1927 [12] carried out research similar to Pfahl [14] and confirmed Pfahl's conclusions.
Niemann 1935 [13] experimented with model clamshells. He investigated the deadweight, the bucket's shape, the soil mechanical properties, the payload and the rope force. Special attention was paid to the width of the grab, leading to the conclusion that the payload is proportional to the width of a grab. The research also led to a confirmation of the work of Pfahl [14] and Ninnelt [12].
Tauber 1959 [17] conducted research on prototype and model grabs. Contrary to Nieman [13] he found that enlarging the grab does not always lead to an increasing payload. The optimum ratio between the grab width and the grab span was found to be in between 0.6 and 0.75.
Torke 1962 [18] studied the closing cycle of a clamshell in sand for three different 39.5 kg model grabs. He first determined the closing path of the buckets experimentally, after which he reconstructed the filling process and the rope forces. His results were promissing, eventhough he did not succeed in predicting the closing curve. An important conclusion reached by Torke [18] is, that the payload is inversely proportional to the cutting angle of the bucket edges. In a closed situation, the cutting angle should be as near to horizontal as possible.
Wilkinson 1963 [19] performed research on different types of grabs and concluded that wide span grabs are more efficient then clamshell grabs. He also concluded that no model laws for grabs exist and that existing grabs are proportioned in about the best way possible. The best grab is a grab that exerts a torque on the soil that is as high as possible especially towards the end of the closing cycle.
Hupe and Schuszter 1965 [6] investigated the influence of the mechanical properties of the soil such as the angle of internal friction. They concluded that grabs intended to handle rough materials like coal should be larger and heavier.
Dietrich 1969 [3] tested a 0.6 m grab and measured the payload for different values of the deadweight, the grab area, the cutting angle and the grain size. He concluded that in hard material 80% of the closing force is used for penetrating the soil, while in soft material this takess only 30% of the force. The width/span ratio should be between 0.6 and 0.7 matching Tauber's [17] conclusions, while the cutting angle should be about 11 to 12 degrees with the horizontal in a closed situation matching Torke's [18] conclusions.
Gebhardt 1972 [4] derived an emperical formulation for the penetration forces in materials with grain sizes from 30 to 50 mm. Grain size and distribution are parameters in the equation, but the mechanical properties of the soil such as the angle of internal friction are absent. He also concludes that a uniform grain distribution results in relatively low penetration forces. Teeth are only useful in rough materials, but they have a negative effect in fine materials with respect to the penetration forces.
Scheffler 1973 [15] made an inventory of grab dimensions and design tendencies in several Eastern European countries. He concludes that most of the grabs are not used to their full potential and also that 80% of the closing force is used for penetration in rough materials confirming the work of Dietrich [3].
Scheffler, Pajer and Kurth 1976 [16] give an overview of the mechanical aspects of several types of grabs. The soil/grab interaction moreover is too simplified or absent. They concluded that after fifty years of research the understanding of grabs is still limited. They refer to Wilkinson [19] as having derived the best conclusions about grab model testing, but regret that prototype results are not available.
Bauerslag 1979 [1] researched the process of grabbing ores of 55 mm with a motor grab. As with Torke [18] he first measured the closing curve (digging path) and then reconstructed the closing process.

From the literature survey it can be concluded, that much research has been carried out in order to find the optimum geometry of clamshells with respect to the payload. The influence of the nature of the bulk material, however, has been underestimated, while no research has been carried out with respect to the use of clamshells under water. Several researchers manage to reconstruct the filling process of a clamshell, once the closing curve is known, but not one of them is able to predict the closing curve. One of the main problems is that grabs are designed by mechanical engineers, while the bulk material taken by the grab often behaves according to the rules of soil mechanics, the field of the civil engineer.
This results in a communications problem. To be able to simulate and thus predict the closing process of clamshells, one needs to study the clamshell operation, kinematics, dynamics (equations of motion) and the soil mechanical behaviour of the material taken. This will lead to a better understanding of the processes involved.

Clamshell grabs as used in dredging industry, consist of six main bodies that can be distinguished as is shown in figure 4. These six bodies are the upper sheave block, the lower sheave block, the two arms and the two buckets. In between the two sheave blocks the closing wire (rope) is reeved with a certain number of parts of line. The hoisting (and lowering) wire is mounted on top of the upper sheave block. A cycle of the grabbing process in a soil which is hard to dig consists of first lowering the clamshell fully opened and placing it on the soil to be excavated. When the clamshell is resting on the soil the hoisting wire is kept slack, so the clamshell will penetrate vertically into the soil by its own weight. This is called the initial penetration. The distance between the to sheave blocks is at a maximum during the initial penetration. Secondly the closing wire is hauled in,
resulting in the two sheave blocks being pulled towards each other and thus causing the closing of the buckets. During this second stage, the hoisting wire is kept slack, so the buckets are allowed to penetrate into the soil.
In soft soils it may be necessary to keep the hoisting wire tight, because otherwise the clamshell might penetrate too deeply into the soil, resulting in a lot of spillage.

Figure 4: The nomenclature of the clamshell buckets.

In this paper, only hard to dig sands will be considered. At the end of the second stage the clamshell is closed and will be raised with the hoisting (and the closing) wire.
Figure 5 shows the stages of the closing cycle of the clamshell. The amount of soil taken by the clamshell depends on the kinematics and the weight distribution of the clamshell and on the mechanical properties of the soil to be dredged.

Figure 5: Three stages of the closing process.

In order to calculate the closing curve of a clamshell, the equations of motion of the moving parts of the clamshell have to be solved. The type of clamshell considered has six main bodies that are subject to motions. These bodies are the upper sheave block, the lower sheave block, the two arms and the two buckets. Because the arms have a small rotational amplitude and
translate vertically with the upper sheave block, they are considered as part of the upper sheave block. The error made by this simplification is negligible.

If a clamshell is considered to be symmetrical with respect to its vertical axis, only the equations of motion of one halve of the clamshell have to be solved. The other half is subject to exactly the same motions, but mirrored with respect to the vertical axis.
Since there are three main bodies left, three equations of motion have to be derived. In these equations weights are considered to be submerged weights and masses are considered to be the sum of the steel masses and the hydromechanical added masses. The weights and the masses as used in the equations of motion are also valid for one half of the clamshell. The positive directions of motions, forces and moments are as depicted in figure 6.

For the upper sheave block the following equation can be derived from the equilibrium of forces:

(1)

The motions of the lower sheave block should satisfy the equilibrium equation of forces according to:

(2)

For the rotation of the bucket the following equilibrium equation of moments around the bucket bearing is valid:

(3)

Figure 6: The parameters involved (forces and moments distinguished in the clamshell model).

As can be seen, equations (1), (2) and (3) form a system of three coupled non-linear equations of motion. Since in practice the motions of a clamshel depend only on the rope speed and the type of soil dredged, the three equations of motion must form a dependent system, with only one degree of freedom.
This means that relations must be found between the motions of the upper sheave block, the lower sheave block and the bucket.

A first relation can be found by expressing the rope force as the summation of all the vertical forces acting on the clamshell, this gives:

(4)

Since there are four degrees of freedom in the equations thus derived:

(5)

One of them has to be choosen as the independent degree of freedom, whilst the other three have to be expressed as a function of the independent degree of freedom. For the independent degree of freedom, f is choosen as the closing angle of the bucket.

To express the motions of the upper and the lower sheave blocks as a function of the bucket rotation, the following method is applied:

The angle of an arm with the vertical a, can be expressed in the closing angle of the bucket by:

(6)

The distance between the upper and the lower sheave blocks can now be determined by:

(7)

As can be seen, the only unknown variable in equations (6) and (7) is the closing angle f. All other variables are constants, depending only on the geometry of the clamshell.
A function h(f) can know be defined, which is the derivative of the distance between the sheave blocks with respect to the closing angle of the buckets.

(8)

If during a small time interval Dt the length of the closing rope l and the closing angle f, are subject to small changes Dl and Df, the change of the vertical position of the upper sheave block Dy_u can be calculated with:

(9)

The change of the vertical position of the lower sheave block Dy_l can be expressed by:

(10)

In equations (9) and (10) i is the number of parts of line.

Dividing the equations (9) and (10) by the time increment Dt gives the equations for the velocities of the upper and the lower sheave block. For the upper sheave block equation (11) is valid.

(11)

The velocity of the lower sheave block can be calculated with:

(12)

The vertical accelerations of the upper and lower sheave block can be calculated by taking the derivative of equations (11) and (12) with respect to the time, this gives for the upper sheave block:

(13)

and for the lower sheave block:

(14)

The vertical acceleration at the centre of gravity of the bucket can be expressed as a function of the vertical acceleration of the lower sheave block and the angular acceleration of the bucket according to:

(15)

The three vertical accelerations can now be expressed as a function of the rotational bucket acceleration. Velocities and motions can be derived by means of integrating the accelerations if boundary conditions are given. The force in the clamshell arm can be calculated from equation (1) if the rope force F_r and the vertical acceleration of the upper sheave block are known.
The vertical cutting force F_cv, the vertical force on the side edges F_ev and the torque on the side edges M_e will be discussed in the next paragraph.
Since the equations of motion are non-linear, the equations have to be solved numerically. The solution of this problem is a time domain solution, in this case using the Newton Rapson iteration method and the teta integration method to prevent numerical oscillations.

The buckets of the clamshell are subject to forces and resulting moments exerted out by the sand on the buckets. The forces and moments can be divided into forces and moments as a result of the cutting forces on the cutting edges of the buckets and forces and moments as a result of the soil pressure and friction on the side edges of the buckets. Figure 6 shows the forces and
moments that will be distiguished in the clamshell model.

The cutting forces on the cutting edges of the buckets can be calculated with the cutting theory of Miedema [7,8] presented at WODCON XII in 1989.
This theory is based on the equilibrium of forces on the layer of sand cut and on the occurrence of pore under pressures. Since the theory has been published extensively, the theory will be summarized with the following equations:

If cavitation does not occur the horizontal force on the cutting edge can be calculated with:

(16)

(17)

If cavitation does occur the horizontal force on the cutting edge can be calculated with:

(18)

For the vertical cutting force:

(19)

The proportionality coefficients c₁, c₂, d₁ and d₂ can be found in Miedema 1987 [7] or 1989 [8].

Figure 7: Typical failure patterns that might occur under deep foundations (ref. 23).

The forces and moments on the side edges were unknown when the research started. At first it was assumed that the forces were negligible when cutting sand. From the model experiments Wittekoek 1991 [21] carried out, it appeared that the computer program CLAMSHELL resulted productions that were too high.
Changing the mechanical properties of the soil within the accuracy range could not solve this problem. Implementing pressure and friction forces on the side edges improved the calculated results drasticaly.

The forces on the side edges are modelled as the forces on strip footings, Lambe & Whitman 1979 [23]. Figure 7 shows some typical failure patterns that might occurr under foundations. The general equation for the pressure force on a strip footing is:

(20)

The friction force on the side surfaces of the buckets can be derived by integrating the shear stress over the side surfaces. It appeared from the research that this part of the forces is negligible in sand.

The coefficients N_c, N_g and N_q can be calculated according to different theories. The best known theory is the theory of Terzaghi for shallow foundations. Theories for shallow and deep foundations have been developed by De Beer, Meyerhof, Brinch Hansen, Caquot-Kerisel, Skempton-Yassin-Gibson, Berantzef, Vesic and Terzaghi. Lambe & Whitman 1979 [23] give an overview of these theories. The different theories mentioned are based on different failure patterns of the soil. All theories are based on drained conditions, meaning that excess pore pressures can dissipate readily. This assumption is reasonable for static foundations, but not for the digging process of clamshells. During the digging process pore underpressures will occurr, increasing the soil pressure on the side edges.
Two problems now occur in modelling the forces on the side edges.
The first problem is, which theory to choose for the side edge forces under drained conditions such as those occurring during the initial penetration and the digging process in dry sand. The second problem involves the modelling of the influence of pore pressures on the side edge forces as it occurs when cutting saturated sand.
The first problem was solved by examining the initial penetration and the digging curves that occurred with 8 tests in dry sand. It required some trial and error to find satisfactory coefficients for equation (20).
The second problem was solved by examining the initial penetration and the measured digging curves in saturated sand. Although the resulting equation for the force on the side edges is empirical, it is based on a combination of Terzaghi's foundation theory and Miedema's cutting theory.

(21)

The pore underpressure Dp in equation (21) follows from the sand cutting theory of Miedema 1987 [7]. The parts of equation (20) containing N_c and N_g appeared to be negligible and thus cannot be found in equation (21).

For the verification and validation of the calculation method as described in the previous paragraphs, a test rig was built in the Laboratory of Soil Movement of the Delft University of Technology.
The test rig consisted of a model clamshell grab, a container filled with 100 mm sand, a vibration device, a cone penetrometer and a data-acquisition system. Figure 8 gives an impression of the test stand. Figure 9 shows the model clamshell used. On the model clamshell two displacement transducers were mounted, to measure the vertical position and the closing angle. In the
closing wire a force transducer was mounted to measure the closing force.
The vibration device was used to compact the sand and thus make it possible to get sand with different soil mechanical properties. The cone penetrometer was used to determine the cone resistance of the sand.

Figure 8: The test rig with the model clamshell grab, a vibration device and a cone penetrometer.

By means of calibration diagrams (Miedema 1987 [7]), when the cone resistance is known, the density, the angle of internal friction, the soil interface friction angle and the permeability of the sand could be determined.
All transducers were connected with the data-acquisition system, so the data could be processed by a computer. The aim of the research was to do tests in dry and saturated sand, compare the results with simulations of the CLAMSHELL program, and adjust the calculation method if necessary. Since the calculation method is fundamental, it should not matter on which scale the
tests are carried out. As explained in the previous paragraph, the forces exerted on the buckets by the sand include a part determined by the mechanical properties of the dry sand and a part determined by the mechanical properties of the saturated sand. Also the forces consist of a part acting on the cutting edges of the buckets and a part acting on the side edges of the
buckets. From Miedema 1987 [7] and 1989 [8] the cutting forces on the cutting edges can be calculated in dry and in saturated sand. What would occur on the side edges was not known when this research started.
To quantify the side edge forces, first 8 tests were carried out in dry sand. Since the force of the closing wire was measured and the real cutting forces could be calculated, the forces on the side edges remained. Repeating this with 14 tests in saturated sand gave a good impression of the influence of saturation on the side edge forces. As a result of these tests, an equation was derived for the side edge forces in dry and in saturated sand as described in the previous paragraph.

Figure 9: Close up of the clamshell model.

Figures 10, 11, 12 and 13 give an example of the test results and the simulations. Figure 10 is the result of a test in dry sand with 10 minutes vibration time. Figure 11 is the result of a simulation with the same mechanical properties of the soil. As can be seen, the digging curves correlate well. The closing force calculated is very smooth, while the closing force measured shows irregularities as a result of the occurrence of discrete shear surfaces in the sand (chipping). The correlation is reasonable however. Figure 12 is the result of a test in saturated sand with 15 minutes vibration time. Figure 13 is the result of a simulation with the same mechanical properties of the soil. Again the digging curves correlate well.
The shape of the simulated closing force as a function of the span differs slightly from the measured shape, but the magnitude of the measured and the calculated closing force correlate well. The angular velocity was derived from the signals of the displacement transducers. The shape of this signal from test and simulation correlates well, although irregularities occur in
the measured angular velocity.

Figure 10: Result of a cutting test in dry sand.

Figure 11: Result of a simulation in dry sand.

Figure 12: Result of a cutting test in saturated sand.

Figure 13: Result of a simulation in saturated sand.

As a result of analyzing the closing process of a clamshell from the point of view of a mechanical engineer and of a civil engineer, a numerical method of calculation has been developed that simulates the closing process very well. The laboratory research carried out has been a great help in adjusting and tuning the computer program CLAMSHELL. The correlation between the test results and the results of the simulations was good. With respect to the mathematical modelling it appears that the forces on the side edges of the buckets are of the same magnitude as the real cutting forces and can certainly not be neglected. With respect to the use of the CLAMSHELL program it can be stated that the program has already been very useful for the prediction of the production of a clamshell used in dredging operations, moreover the program can also be of great help in designing improved clamshells as well. Studies have already been carried out by Great Lakes, to find optimum clamshell kinematics and mass distribution. A next step in this research will be, the verification and validation of clay cutting with clamshell grabs.

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23. Lambe, T.W. & Whitman, R.V., "Soil Mechanics". John Wiley & Sons, New York 1979.

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