In order to minimize material waste or borrow, it is necessary to produce what is called a Mass Haul diagram. This is essentially a plot of cumulative volume of soil against distance along the road, often called the chainage. Cut volumes are taken to be positive and fill volumes to be negative.
Calculation of Cross Sectional Area
The first stage in the production of the Mass Haul Diagram is the calculation of the cross sectional areas of cut or fill at different points along the road.
For a cut or fill on horizontal ground.
Figure 1 – Typical Cut Cross Section |
Assuming a cut such as the one above, the cross sectional area is given by:
Area = h.2b + 2nh²/2 = h(2b + nh)
For a cut or fill on sloping ground
Figure 2 – Typical Sloping Cut Cross Section |
Assuming a cut such as the one above, the cross sectional area is found firstly by calculating W_{L} and W_{G}:
W_{L} = S(b+nh)/(S+n)
W_{G} = S(b+nh)/(S-n)
Thus Area = ½(h + b/n)(W_{L} + W_{G}) – b²/n
For more complicated cross sections, simply combine the above. It should be noted that this is NOT part of the design process for the slope stability.
Cumulative Volumes
Once the cross sectional areas are known at various points along the road, it is possible to calculate the cumulative volume along the cut by interpolating between the different points.
The simplest way of doing this is to assume a straight line variation and use the prismatic rule. Other slightly more complicated methods involve using Simpsons rule or similar. Do not forget to take account of the bulking factor or shrinkage factor although care should be taken not to use them both as this will produce incorrect results. If you are using the shrinkage factor then changes in volume due to excavation is accounted for automatically. The same is true for the bulking factor.
EARTHWORK FORMULAS
Area | |
Area by Coordinates | |
Area = [X_{A}(Y_{B} – Y_{N}) + X_{B}(Y_{C} – Y_{A}) + X_{C}(Y_{D} – Y_{B}) + …+ X_{N}(Y_{A} – Y_{N-1})]/2 | |
Trapezoidal Rule | |
Area = w[(h_{1} + h_{n})/2 + h_{2} + h_{3} + h_{4} + …. + h_{n-1}] w = common interval length | |
Simpson’s 1/3 Rule | |
w = common interval length | |
Volume | |
Average End Area Formula | |
V = L(A_{1} + A_{2})/2 | |
Prismoidal Formula | |
V = L(A_{1} + 4A_{m} + A_{2})/6 | |
Pyramid or Cone | |
V = h(area of base)/3 |
For a more accurate measurement of dike volume on rough ground, you should apply the following formula, known as Simpson’s rule, where: V = (d ÷ 3) x [A_{1} + A_{n} + 4(A_{2} + A_{4} + … A_{n-1}) + 2(A_{3} + A_{5} + … A_{n-2})]. Proceed as follows:
(a) Divide the length of the dike into an odd number n of cross-sections at equal intervals of d metres.
(b) Calculate the area A of each cross-section as explained earlier.
(c) Introduce these values into the above formula