INTRODUCTION TO VOLUME MEASUREMENTS Volume measurements are needed for three different categories of pay items:

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1 INTRODUCTION TO VOLUME MEASUREMENTS Volume measurements are needed for three different categories of pay items: Earthwork --items such as borrow excavation, and subsoil excavation Concrete -- the various classes of concrete used in bridges and other structures; and Truck Measurements -- material delivered by trucks and measured by volume. Each of these categories is handled differently in the field. UNITS OF MEASUREMENTS The pay item unit of measurement for volume usually is the cubic yard. Keep this relationship in mind: One cubic foot = 1728 cubic inches; One cubic yard = 27 cubic feet; and One cubic foot = 7.48 gallons. METHODS FOR COMPUTING VOLUMES Volumes of earthwork usually are computed from cross sections taken before and after construction. The volume of concrete in structures is most often computed by formulas for the geometric shapes involved. And, when items are paid for by truck quantity, use the manufacturer s certification or permanent decal showing the truck s capacity and then simply count the number of loads delivered. CROSS SECTIONS In Chapter Two we talked about cross section notes and how they shall be recorded in field notebooks. In this chapter we want to learn how to use cross sections for measuring volumes of earthwork.

2 Figure 4-1 below, illustrates typical earthwork cross sections. The example shows all fill sections probably roadway embankment. Sometimes the sections may be all cut, such as borrow areas or ditches. Quite often, however, both cut and fill areas are shown on the same section. A common method of determining volumes from cross sections is that of average end areas. It assumes that the volume between successive cross sections is the average of their end areas multiplied by the distance between them. This is expressed in the formula: V (Ft. 3 ) = [A1 (FT 2 ) + A2 (Ft 2 )] X L (Ft) 2 In which V is the volume in cubic feet, A1 and A2 are the end areas in square feet of successive sections and L is the length in feet between the sections. (The end areas will be divided by 2 to come up with the average end area). This formula is exact when A1 equals A2 in both area and configuration, but is only approximate when the cross sections have different areas. It is generally accepted as close enough for computing earthwork volumes. Later in this chapter we will look at a more accurate approach called the prismoidal formula. It is used for computing volumes of concrete, a much more expensive item. Using the examples of cross sections shown in Figure 4-1, let's see if we can compute the volume of earthwork by applying the above formula. The end areas shown on the sections were computed by a method described in the Volume Measurements section. We must compute the volume between each pair of cross sections and add the individual volumes to obtain the total volume of earthwork between station (9 + 75) and station ( ). See Figure 4-2 below.

3 FIGURE 4-1 Begin Sta Area = 0 End Sta Area 7 = 0 To make the answer come out in cubic yards, we must divide by 27. (Remember that the end areas were divided by 2 originally). So, the formula will now read: (Below are two equations of how to achieve this).

4 Volume (V) = (A1 + A2) L or V = (A1 + A2) L 2 X Figure VOLUME MEASUREMENTS 1. Volume measurements are needed for which three different categories of pay items? A. Slope Pavement, Pile Driving and Fencing. B. Concrete, Truck measurement and Earthwork. C. Concrete, Pile Driving and Earthwork.

5 D. Fencing, Truck Measurement and Slope Pavement. E. None of the above. 2. How many Cubic Feet are in 350 Gallons? Note: 1 CF = 7.48 Gallons. Round your answer to the tenth of a cubic foot. A cubic feet B cubic feet C cubic feet D cubic feet E. None of the above. 3. Based on the areas determined for the Stations shown below, what is the total volume of earthwork between Station and 72+75? (solve to the nearest CY) A. 1,406 cubic yards B. 2,828 cubic yards C. 1,916 cubic yards D. 1,667 cubic yards Station Area Volume Ft Ft Ft Ft Ft 2 Total

6 4. The table shows the end areas determined for the indicated cross sections. Compute the total volume of earthwork between stations and (Answer to the nearest Cubic Yard) A. 2,925 Cubic Yards B. 2,332 Cubic Yards C. 1,927 Cubic Yards D. 1,513 Cubic Yards CURVATURE CORRECTION Station Area (S.F.) Volume (CY) Total = When volumes are computed along a curved base survey line, some error will be introduced unless the center of gravity of the cross sectional area lies approximately along the base line. This situation is illustrated in the example below: G = Center of Gravity R = Radius of base line

7 Note: Multiline handles the curvature correction calculations. VOLUME FORMULAS Cross sections do not work well for computing the volumes of some pay items such as reinforced concrete and excavation. For these items, it is much better to measure the dimensions of the construction and use conventional formulas to compute geometric shapes. The Construction Mathematics training course provides a good background in the use of formulas for calculating volumes. Let's take a quick look at some of the formulas we will be using. The simplest geometric shape of course, is the rectangular solid with opposite sides parallel.

8 In both cases, V =LWH (volume equals length times width times height) Example: Find the volume of the concrete block seen here. First, we must make sure all measurements have the same components. The width is 2 Ft and 3 inches. The inches will have to be converted into feet = 0.25 Ft. The Width is 2' 3" = 2.25 Ft. V = LWH V = 8 X 2.25 X 5 V = 90 Cubic Foot. If the answer is in Cubic Yards, then we must divide by 27 Ft/CY, and the Volume will be: V = = 3.33 CY = 3 CY (If answer is to be to the nearest CY) When you studied areas, you learned that a triangle is half of a rectangle or parallelogram. The same concept applies to volumes.

9 Example: Calculate the volume of the triangle seen below to the nearest cubic foot. Now how about trapezoidal solids? Do these formulas look familiar? When you stop to think about it, we are really computing an end area and then multiplying by a third dimension to find the volume. Example: Calculate the volume of the trapezoid seen below to the nearest cubic foot.

10 First, we must make sure all measurements have the same components (inches need to be converted to feet). a = 4' 4" and 4" 12 = 0.33; a= 4.33 Ft. b = 9' 4" and 4" 12 = 0.33; b = 9.33 Ft. H = 7' 2" and 2" 12 = 0.17; H = 7.17 Ft. W= 3' 9" and 9" 12 = 0.75 W= 3.75 Ft V = [(a + b) 2] HW = [( ) 2] X 7.17 X 3.75 = V = = 184 CF Let s try it with a cylindrical solid and a cone: Why do we divide by 3 instead of 2 in the cone formula? The tip of the cone obviously has a zero area, and if we average the end areas the formula should be: V = A x H 2 But this is wrong. Remember, when we talked about cross sections we said that the average-end-area method was close enough for earthwork volumes but not accurate enough for items like concrete. So, we need to use the cone formula or "prismoidal formula." The prismoidal formula is: Where:

11 A1 and A2 are the two end areas and Am (m = mean) is the area at a point halfway between the two ends. Let us solve some examples together: Example 1 Calculate the volume of the cone using the cone formula. Calculate the answer to the nearest cubic foot. H = 30 in V = πr 2 H 3 Note: the answer is to the nearest cubic foot We will first convert the inches to feet = 2.5 Ft = 0.83 Ft. Example 2 R =10 in V = X (0.83) 2 X V = 1.8 = 2 CF How about an example with a cylinder? D= 3 ' Calculate the volume of the Cylinder to the nearest Cubic Yard. H= 8' V = πr 2 X H R = (D 2) = 3 2 = 1.5' V = X (1.5) 2 X 8 R = 6' V = 56.5 CF 27 CY/CF = 2.1 = 2 CY

12 All right, so when should the prismoidal formula be used? Let's look at a definition and see what it means to us: Prismoidal shape: A solid with parallel end areas and sloping sides, but whose mid-area (mean area) is not necessarily the average of the two end areas. How and when we apply the prismoidal formula to our work is straightforward. Since we usually calculate from vertical plane to vertical plane (i.e., cross sectional areas) the "end areas" in the above definition are self-explanatory. All that is left then is to visualize the sides, and how they slope in relationship to one another. It is when the side planes taper in or out in relationship to one another that the average-end-area method is not accurate enough for the more critical pay items. Take the figure below. Both end areas are zero, and yet the figure does have volume. By using the prismoidal formula, the figure's volume can be computed accurately: You will see many applications of prismoidal formulas in your work, especially in retaining wall, truck body and concrete structure calculations. As you saw earlier, the average-end-area method does not apply accurately to computing the volumes of cones. So then, how are the cone formula and the prismoidal formula related? This is explained next.

13 The prismoidal formula comes up with the correct answer for the volume of cones -- and it should always be used for concrete volume computations when the average of the end areas is not the same as the mean area. Note: In this course, please use the cone formula volume for the cone examples; V = πr² x H 3 And for the odd shape forms that have an average mean use the prismoidal formula: Let s solve an example of an odd shaped concrete block with an average area: See example below:

14 Calculate the volume for the above odd shape of concrete using the prismoidal formula. Make the answer to the nearest cubic yard. V = (A1 + A2 + 4 Am) H 6 A1 = 0, A2 = 0 The middle area = 4 (area of the square), and 4' 9" = 4.75'; the Height (H) = 21' 3" = 21.25' Total Volume = 4 (4.75 X 4.75) X = CF 27 CY/CF = = 12 CY 6 1. Use the volume formulas we ve been discussing to compute the volumes of the shape to the nearest cubic Yard. A. 784 Cubic Yards B. 896 Cubic Yards C. 671 Cubic Yards D. 1,009 Cubic Yards E. None of the above

15 2. Use the volume formula we ve been discussing to compute the volumes of the shape to the nearest tenth of a cubic yard. A Cubic Yard B. 0.9 Cubic Yard C. 1.3 Cubic Yard D. 2.8 Cubic Yard E. None of the above.

16 3. Use the volume formulas we ve been discussing to compute the volumes of the shape to the nearest cubic yard. A. 1,795 Cubic Yards B. 1,884 Cubic Yards C. 1,978 Cubic Yards D. 2,077 Cubic Yards 4. Use the volume formulas we ve been discussing to compute the volumes of the shape to the nearest tenth of a cubic yard. A Cubic Yards B Cubic Yards C Cubic Yards D Cubic Yards E. None of the above.

17 5. Use the volume formulas we ve been discussing to compute the volumes of the shape to the nearest tenth of a cubic yard. A Cubic Yards B Cubic Yards C Cubic Yards D Cubic Yards E. None of the above

18 6. The prismoidal formula should always be used for concrete volume computations when the mid-area (mean area): A. Is less than 50 square feet B. Is equal to the average end areas. C. Is not necessarily the average of the two end areas D. Is greater than 50 square feet. E. None of the above 7. The prismodal formula is A. True B. False

19 SUBSOIL EXCAVATION Where muck, rock, clay or other material within the limits of the roadway is unsuitable it must be excavated. The area must then be backfilled with suitable material. The volume of subsoil excavation is measured and computed with cross sections. But you must be aware of a few points that make these measurements different from those for regular excavation. On the basis of preliminary soil investigations, the plans normally show the limits of proposed subsoil excavation. Measurement and payment are limited to these lines unless otherwise authorized by the engineer. The authorized limits, as staked in the field, must be recorded in the earthwork notes for each pocket of muck excavation. If during the excavation the muck is found to be deeper than plan depth, the notes shall be corrected by striking through the original limits and writing the new authorized limits. The corrected notes shall also be dated and initialed. Figure 4-3* and Figure 4-4* shows an example of these field book entries. To see Figure 4-3 and 4-4, please click on their links respectively. Using these notes, the volume is computed in the same manner as for regular excavation except for the following special considerations: 1. Payment cannot be made for excavation outside the authorized limits. 2. All authorized work outside the original limits must be documented by the Project Administrator. 3. Authorized subsoil excavation made to a depth of 0 to 5 feet below the depth shown on the contract plans will be paid for at the unit price bid. 4. Authorized excavation to a depth of 5 to 15 feet deeper than shown on the plans will be paid for at the unit price bid plus 25 percent of the unit price. 5. Additional authorized excavation, more than 15 feet below plan depth, will be considered a change in character of the work and will be paid for as extra work. Note: Items 3, 4 and 5 are according to current specifications. Figure 4-5 and Figure 4-6 (below) illustrate these special situations. Separate computations and payment must be made for volume on each of the three layers of subsoil excavation:

20 1. Subsoil excavation at unit price bid; 2. Extra depth subsoil excavation at unit price plus 25%; and 3. Extra depth subsoil excavation paid for as extra work. The end areas of each layer can be computed by the latitude and departure method, using survey cross sections. The volume of each layer is then computed by the average-end-area method the same as for any other earthwork computation. Study the next two figures: Figure 4-5

21 CASE I: In every case, the point of reference for determining extra depth subsoil excavation should be the lowest elevation that muck is shown on the plan cross section and this elevation projected horizontally across the entire cross section in making this determination. Each cross section shall be considered separately. Figure 4-6 CASE II: When subsoil excavation is encountered, where none was shown on the plans, in separate areas, or as an extension to areas that were shown the point of reference shall be the lowest elevation on the original ground line or the lowest elevation of the plan excavation whichever is lower.

22 1. Determine the volume of the figure below to the nearest cubic foot. A. 129 Cubic Feet B. 117 Cubic Feet C. 136 Cubic Feet D. 157 Cubic Feet E. None of the above

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