3.8: Finding the Shear Angle

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The unknown in all the mechanisms is the shear angle β. With the assumption that nature will choose the mechanism configuration that requires the least energy and this energy equals the horizontal force F_h times the cutting velocity vc times time, the shear angle β should be chosen where the horizontal force F_h is at a minimum. In some cases an analytical solution exists by taking the derivative of the horizontal force F_h with respect to the shear angle β and making it equal to zero. The second derivative has to be positive in this case. In other cases it is more convenient to determine the minimum numerically. This minimum value depends strongly on the blade angle α and the blade height – layer thickness ratio h_b/h_i. This minimum also depends strongly on the soil properties and thus the type of soil. Different soils will have shear angles in a different range. Different cutting mechanisms will also have shear angles in different ranges. For saturated sand with blade angles α from 30° to 60°, the shear angle β will range from 30° to 20°. For clay, the shear angle depends strongly on the ratio of the adhesion to the cohesion. For very strong clays with a low relative adhesion the shear angle can be in the range of 60° to 75° for blade angles α from 30° to 60°. For soft clays with a high relative adhesion the shear angle is much smaller, from 30° to 40°. In general one can say that the shear angle decreases with increasing blade angle, internal/external friction angle and adhesion.

The criterion of least energy is arbitrary but reasonable. Other criteria may be applied to find the shear angle. Also other mechanisms may be applied leading to slightly different shear angles. In this book it is assumed that the shear plane is a straight line, which is questionable. First of all, the shear plane does not have to be a line without thickness. An area with a certain thickness is also possible. Secondly, the shape of the shear plane could be curved, like a circle segment. The advantage of the approach chosen is, that one can compare the different mechanisms more easily and the models derived give more insight in the basic parameters.