Mass Calculation - Cross Section Method

Calculation Method

The basis for this method is cross-sections along a line definition or horizontal curvature in an SFI model.

A cross-section can contain physical layers generated from terrain strata, theoretical layers built from the SFI model, or sections through 3D objects.

Figure: A = Physical layers, B = Theoretical layer

With the cross-section method, we can calculate both areas and volumes. This is described in detail below.

Area Calculation

We have the following area calculation types:

• Horizontal Area
• Vertical Area
• Inclined Area

These calculation types compute areas for each profile with the lengths that apply in this profile over a length equal to the profile interval.

Figure: L = Length, P = Profile distance

Exception for first and last profile

The first profile is calculated from the starting point to ½ interval past the profile value. The last is calculated from ½ interval before the profile to the specified end point.

How does the program calculate the lengths?

This is best illustrated with an example. In this example, we have two layers (1 and 2) that cross each other.

Figure: 1 = Layer 1, 2 = Layer 2

For Horizontal Area and Vertical Area, the maximum extent of the layers determines the lengths:

• Horizontal Area:
  
  Figure: 1 = Layer 1, 2= Layer 2, Green line = Horizontal Area
• Vertical Area:
  
  Figure: 1 = Layer 1, 2 = Layer 2, Green line = Vertical Area

Inclined Area

For inclined area, the calculation is different. In cases where layers cross each other, the layers will be connected as shown in the example below:

Volume Calculation

We have the following volume calculation types:

• Volume
• Volume Difference
• Volume Intersection
• Volume Union

All of these calculation types use the standard method for mass calculation in cross-sections. Masses are calculated for each profile with the areas that apply in this profile over a length equal to the profile interval.

Exception for first and last profile

The first profile's mass is calculated from the starting point to ½ interval past the profile value. The last is calculated from ½ interval before the profile to the specified end point.

For all parts of the cross-section, area and area center of gravity are calculated. When areas are converted to masses, the profile distance at the area's center of gravity is used. This accounts for the "pie slice" effect in curves.

0-profiles

In a zero profile, the area equals 0. The volume then becomes half as shown in the figure below.

Figure: A = Area, B = 0-profile

Now let's look at how the program calculates the area for volume calculation. This depends on the calculation type.

Calculation Type: Volume

The method for calculating areas in the cross-section with From layer and To layer uses polygon geometry functions. Rules for calculation between from-layers and to-layers are set up. Any number of from-layers and to-layers can be specified. The same polygon is used for calculation and documentation, What You See Is What You Get.

Example: Calculation of soil cutting in cross-section

Involved layers:

• Trench
• Soil
• Rock

Figure: A = Soil layer, B = Rock layer, C = Trench

Each layer defines a polygon.

Figure: Blue = Polygon of the trench layer. A = Soil layer, B = Rock layer, C = Trench

From polygon:

The From polygon is created by taking the "union" between all from-layers (i.e., the uppermost of the from-layers).

Figure: From polygon in red when rock and trench layers (two from layers) are selected as From layer. A = Soil layer, B = Rock layer, C = Trench

From-polygon

The "From polygon" is created by taking the "union" between all from-layers (i.e., the uppermost of the from-layers).

To polygon:

The To polygon is created by taking the "intersect" between all to-layers (i.e., the lowermost of the to-layers).

Figure: To polygon in red when only soil is selected as to-layer. A = Soil layer, B = Rock layer, C = Trench

The Soil cutting polygon is obtained by "Subtracting" the "From polygon" from the "To polygon" (i.e., the difference between to-layer and from-layer).

Figure: \(To polygon - From polygon = Result\). A = Soil layer, B = Rock layer, C = Trench

Calculation Type: Volume Difference, Volume Intersection, and Volume Union

For the calculation types Volume Difference, Volume Intersection, and Volume Union, we add layers that together form a polygon. We must add layers for Polygon1 and Polygon2. When both polygons are defined, the program calculates the result polygon. Below are principle sketches for the different calculation types.

Difference:

Figure: Volume difference

Intersection:

Figure: Volume intersection

Union:

Figure: Volume union

How are polygons formed?

A polygon is formed from a list of layers in the following way:

1. Start with layer 1 (the topmost) in the list
2. Go from start point to end point in layer 1
3. Continue with layer 2 in the list
4. The end of layer 1 is connected to the start point in layer 2
5. Go from start point to end point in layer 2
6. And so on.

Important about layer order

The order of layers in the list is important if we have more than two layers. The layers should follow each other, and the positive direction is clockwise. Layers with negative direction must be reversed in the mass type. This is set under advanced settings for the layer, and the selection is marked with a dash (-).

Correction for "Pie Slice Effect"

Figure: Pie slice effect

The result (as it appears in the horizontal list field and report) is given by:

\(Volume = Cross-section × LP × factor\)

\(Factor = La / Lp\)

\(La = Lp × (R - Arm) / R\)

The length (LP) of the calculation and the curve correction (factor) can also be found in the horizontal list field.

Correction for Vertical Curvature

When working with tunnels, it's possible to tilt the profiles so they stand perpendicular to the vertical curvature. For regular road projects, however, it's common to generate with vertical profiles. The error becomes so small that we choose to ignore it. The option for tilting cross-sections is set in the profile generator.

Display of vertical correction

If we activate the option to tilt profiles, the vertical correction (i.e., the ratio between horizontal and inclined distance) will be shown in a separate column in the horizontal list field.

That the profiles are generated with tilting can be seen in the vertical list field for cross-section editing. For tilted profiles, we also get information about the inclination and height of the CL in the profile.

How does tilting affect mass calculation?

Some examples will illustrate this.

Example of projected blasting mass

In this example, the red line (dashed) illustrates the theoretical blasting contour.

Figure: Horizontal mass calculation, without correction for vertical curvature. H = Horizontal distance

Figure: Tilted mass calculation, with correction for vertical curvature. S = Inclined distance (m)

In this case, we see that tilting affects the result.

Example of measured blasting mass

In this case, we have measured the rock after blasting (green dashed line).

Figure: Horizontal mass calculation, without correction for vertical curvature. H = Horizontal distance

Figure: Tilted mass calculation, with correction for vertical curvature. S = Inclined distance (m)

In this case, we see that tilting does not affect the result.

Conclusion

Recommendations for tilting

1. If we are designing a tunnel, we must tilt the cross-sections to get correct projected masses.
2. If we scan an existing tunnel and want to document completed masses, the result will be the same whether we tilt or not.