Calculation of the failure wedge, B
The 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 has to be calculated to calculate the friction of the soil along the fracture lines C, as shown in the free body diagram.
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; the analytical approach is given here for those who are interested in this approach.
Numerical method
approach
The 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.
Parameters
The geometry of the failure wedge depends on the following four parameters:
Geometry of the anchor
Position of the anchor in the soil
Internal angle of friction of the soil, φ
Pulling force on the anchor
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 blade
An 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 force
Secondly 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:
The
magnitude of the pulling force, E
The fluke/shank angle, α
The penetration angle, β
This correction is shown in the figure below.
Volume calculation
The 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
Asegment: 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
The side view of all anchors is the same and shown in the figure below. From this side view we may conclude that the height of the different cross sections is not depending of the geometry of the anchor blade and only depending of the height of the centre of gravity of the anchor, the length of the anchor, the angle of internal friction and the correction angles, which are all numbered and also shown in the figure below.
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 hulpwaarden Htop and Hbottom can be calculated with the following formulas:
The height of each segment can now be determined by the following formulas, in which x is zero at the centre of gravity of the anchor and positive in the direction of the pulling force (to the right in the figures):
if
else
A correction should be included for the aft end of the anchor (left in the figures), where the soil above the anchor blade does not take part of the failure edge.
Now the height of every segment is calculated 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
In the formulas above the following parameters are used:
:
Height of the centre of gravity of a rectangle, which can be
determined with:
:
Breadth of the centre of gravity of a rectangle, which can be
determined with:
: Surface area of the segment
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 overhere, 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 tripple 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 rightern figure above there is a certain overlap area between the boundary lines based on the line of symmetry in the middle of the anchorblade 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 anchorblade its downward points will be cut off by the anchorblade and the segment will have a shape like this:
In this case the anchorblade should be added as a boundary of the segment and only rectangles which lie above the anchorblade 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.