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Calculation of the failure wedgeThe goal of this sub model is to calculate the amount of soil of the failure wedge B, as shown in the free body diagram. Besides the amount of soil in the failure wedge the surface area at the outside of the failure wedge will be calculated, based on which the friction of the soil along the fracture lines C, as shown in the free body diagram, can be calculated.
The amount of soil in the failure wedge can either be calculated
analytical or numerical. The analytical approach however can only be
applied to anchors with a very simple shape, like rectangles or
triangles. The numerical approach on the other hand provides a
general algorithm to calculate the amount of soil of the failure
wedge and its surface area and can be applied to more complex anchor
blade shapes. Since we are interested in a general solution we will
focus on the numerical approach. ApproachThe provided method to calculate the amount of soil in the failure wedge will be presented step by step: Starting at the beginning of the problem with explaining which parameters play a part in the calculation process and going more and more into detail to a final numerical method to calculate the amount of soil in the failure wedge and its surface area. First the volume of the failure wedge will be determined. When this is done the surface area of the failure wedge can easily be calculated using the same approach and formulas. At the end of the description of the calculation method some critical remarks will be given for this method. ParametersThe geometry of the failure wedge depends on the following four parameters:
Since the geometry of the anchor is already discussed here we will start the analysis of the failure wedge with the position of the anchor in the soil and add the geometry of the blade later. Position of the anchor bladeAn anchor shall not lay horizontally in a soil until it reaches its final position, but will be rotated by an certain penetration angle β, as is shown by the anchor on the right in the picture below.
As a result of this rotation the total failure wedge will be larger than when the anchor blade was in its initial horizontal position at the same depth of the centre of gravity H. This is because the wedge at the right side has increased more than the wedge at the left side has decreased. Pulling forceSecondly the pulling force at the shank of the anchor will be taken into account. As
a result of the pulling force the angles of the failure wedge will
not be the same any more as those of the internal angle of friction
of the ground. A correction angle on the topside of the anchor as
well as on the lower side of the anchor has to be taken into account.
These correction angles are dependent of:
This correction is shown in the figure below.
Volume calculationThe
amount of soil in the failure wedge can be calculated when the
failure wedge is divided in segments of a certain thickness in the
direction of the pulling force, as is shown in the figure below.
When
the surface area of each of these segments is known the total amount
of soil can be calculated. This is the sum of the areas of all
segments multiplied by their thicknesses; in formula form this is:
In which: n: Number of segments Aseg: Surface area of an individual segment Δt: Thickness of the segment This numerical method is visualized by the triangle above and the more complex double triangle below.
The
shape and the size of each of the surface areas as shown in the two
pictures above has now to be determined and calculated. This can be
done based on the heights of the cross sections, the internal angle
of friction of the soil and the geometry of the anchor blade.
Height
Although the picture above looks a bit complex, we can easily calculate the height of a segment for an anchor of arbitrary shape from it. The numbers in the picture represent a certain combination of angles, which are: 1: β 2: φ 3top: φ-γtop 3bottom: φ+γbottom 4top: 90-(φ-γtop) 4bottom: 90-(φ+γbottom) Based on the penetration angle β and the length of the anchor the heights Htop and Hbottom can be calculated with the following formulas:
if then
Now the height of every segment is determined the surface area per segment should be calculated. This is off coarse simple to calculate analytical for a triangular wedge, but since we want to derive a general algorithm for an arbitrary anchor blade shape a numerical method should be used. Each segment will have a different shape and size depending on the position of the segment and the geometry of the anchor blade. To calculate the surface area of each segment little rectangles, with a width Δb and height Δh, will be positioned from the middle of the segment to one of its sides. It will be checked if each square lies within the boundaries of the segment. If this is true than the surface area of the rectangle (Δb*Δh) will be added to the surface area of the segment; if the rectangle lies not within the boundaries of the segment its area will not be added to the surface area of the segment. When the position of a rectangle is beyond B the checkingprocess will start again for the next row above the other; this can not be allready done when the position of the rectangle is beyond the boundary of the segment for reasons which will become clear later. Since anchor blades are symmetrical, the failure wedge and thus the segments will also be symmetrical, so it is sufficient to calculate only half of the surface area of a segment and multiply this area times two. The process as explained above is visualized in the figure below for a triangular segment.
In this figure Δh is H divided by a certain integer N_h, Δb is B divided by an integer N_b and B = H.tanφ. For a triangle it is easy to check if the centre of gravity of a square lies within the boundary of the triangle and to calculate the surface area of the segment. This can be done with the following mathematics: for if then the rectangle lies within the boundaries and
The example above, by which the method is explained, does only account for a triangular segment, which boundaries are defined by H.tan(φ). For more complex segment shapes the boundaries have to be determined with a more general method. The boundary of an arbitrary segment can be determined by taking the lines of symmetry of the triangles by which the anchor is built up as a starting point. Since all anchors are defined as combinations of triangles, as is explained here, the lines of symmetry of these triangles is also defined. From these lines of symmetry the boundaries of a segment are defined by B = H.tanφ. Depending on the position of the lines of symmetry of the triangles of the anchor the formula can be applied in one or in two directions from the line of symmetry: it will be applied only in one direction if the line of symmetry of the triangle of the anchor lies on the line of symmetry of the segment; it will be applied in two directions if the line of symmetry of the triangle of the anchor is off the line of symmetry of the segment. The figures below visualize this approach for one, two and triple triangular anchor shapes (the boundaries of even more complex anchors can also be determined with this method in the same way as is done for these examples).
As can be seen in the most righter figure above there is a certain overlap area between the boundary lines based on the line of symmetry in the middle of the anchor blade and the the boundary lines based on the lines of symmetry at the sides of the blade. This overlap however does not influence the surface area which has to be determined, since the rectangles, which will built up the total surface area, still lie within the boundaries of the total surface area. The figures above also show why it is important to position the rectangles all the way to the total breadth of the segment: it is possible that certain parts of the failure wedge lie behind a 'gap' in the failure wedge. These parts would not have been taken into account if the algorithm would have stopped when the boundary was reached. The boundary of the segment is now defined by the main line of symmetry of the anchor and the boundary lines based on the height of the segment and φ. However, when a segment is positioned above the anchor blade its downward points will be cut off by the anchor blade and the segment will have a shape like this:
In this case the anchor blade should be added as a boundary of the segment and only rectangles which lie above the anchor blade will be added to the surface area of the segment. This can be included in the calculations with the following mathematics:
The volume of each segment can now be calculated based on the surface are of each segment and its length Δl. The total volume of the failure wedge is the sum of the volumes of all segments.
Surface area of the failure wedgeTo calculate the friction of the soil along the fracture lines C, as shown in the free body diagram, the surface area of the failure wedge needs to be calculated. This area can be determined with the same method as has been used to determine the boundary per segment, but now not the position of the boundary has to be know but its length. The length of the boundary for each segment will be calculated analytical based on the height of each segment and the number of triangles by which the anchor is built up. The calculation method of the length of boundary of each segment will be explained by means of the pictures below.
To determine the boundary length per segment it should first be determined if there is any overlap between the right and the left side of the line of symmetry of the segment. This can be done with the following statement: if then there is overlap between the right and the left side of the line of symmetry of the segment. If this overlap exists a correction width Bcor has to be defined, which is: This formula can be applied general as is demonstrated in the second and the third of the figures above: Scor in the third figure can be is calculated by:
The length of the boundary at the outer side of the segment can be easily be calculated by:
In case of overlap between the right and the left side of the line of symmetry of the segment the surface area of a segment becomes:
At the aft end of the anchor the upper horizontal surface of the segment should be included, since this is where the top of the segment lies below the seabed. How this can be done in the model is illustrated by the picture below.
If
Critical remarksThe results of the method as described above are suitable to be used for the calculation of the holding capacity of the anchor. However a few critical remarks can be made which can result in more accurate results or by which the method can be improved. – The method is based on the geometry of the horizontal part of the flukes only, the vertical endplate of the anchor is not taken into account. It is assumed that the influence of the vertical endplate on the failure wedge is small end can be neglected. This neglection results in a smaller volume of the failure wedge, which is a conservative approach of the calculation. – The failure wedge as determined above is built up out of geometrical shapes, with sharp edges and corners. In reality the edges and corners of the failure wedge will be more rounded and the volume and surface area of the failure wedge will be larger. Again the method as described above results in conservative solutions. – Since this calculation method is a numerical method the volumes and surface areas are determined discrete and not continious. This will result in small errors, which can be minimized when Δl, Δb and Δh are decreased. |
Copyright © 2005 Project group 1
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