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**Measurement of Volume of Earth work from Cross-Sections: **

The length of the project along the centre line is divided into a series of solids known as prismoids by the planes of cross-sections. The spacing of the sections should depend upon the character of ground and the accuracy required in measurement.

They are generally run at 20m or 30m intervals, but sections should also be taken at points of change from cutting to filling, if these are known, and at places where a marked change of slop occurs either longitudinally or transversely.

The areas of the cross-sections which have been taken are first calculated and the volumes of the prismoids between successive cross- sections are then obtained by using the Trapezoidal formula or the prismoidal formula. The former is used in the preliminary estimates and for ordinary results, while the latter is employed in the final estimates and for precise results.

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The prismoidal formula can be used directly or indirectly. In the indirect method, the volume is firstly calculated by trapezoidal formula and the prisomoial correction is then applied to this volume so that the corrected volume is equal to that as if it has been calculated by applying the prisomoidal formula directly. The indirect method being simpler is more commonly used.

When the centre line of the project is curved in plan, the effect of curvature is also taken into account specially in final estimates of earthwork where much accuracy is needed. It is the common practice to calculate the volumes as straight as mentioned above and then to apply the correction for curvature to them.

Another method of finding curved volumes is to apply the correction for curvature to the areas of cross-sections, and then to compute the required volumes from the corrected areas from prismoidal formula.

**Formulae for Areas of Cross-Sections:**

**The following are the various cross-sections usually met with whose areas are to be computed:**

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1. Level section.

2. Two-level section.

3. Side-hill two-level section.

4. Three-level section.

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5. Multi-level section.

**Notations., Refer fig. 12.1: **

**Let: **

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b = the breadth of formation or sub-grade which is usually constant.

S: 1 = the side slope (S horizontal to 1 vertical).

1 in r = the transverse slope of the original ground (1 vertical in r horizontal)

h = the height of earthwork (cutting or filling) on the centre line

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h_{1} and h_{2} = the side heights, i.e. the vertical distances from formation level to the intersections of the side slopes with the original surface.

W_{1} and W_{2} = the side-widths or half breadths i.e. the horizontal distances from the centre line to the intersections of the side slopes with the original surface.

A = the area of cross-section.

Formulae for the dimensions of the cross-sections of cutting and filling for the above cases are given below and should be verified by the readers as exercises.

**1. Level-Section (Fig. 12.2):**

In this case the ground is level transversely.

**2. Two-Level Section (Fig. 12.1):**

In this case, the ground is sloping transversely, but the slope of the ground does not intersect the formation level.

**Side-Hill Two-Level Section (Fig. 12.3): **

In this case the ground is sloping transversely, but the slope of the ground intersects the formation level such that one portion of the area is in cutting and the other in filling (part cut and part fill).

**Note:**

When filling extends beyond the centre line i.e. when the area in filling is greater than that in cutting, the equations 12.3 and 12.4 are used for finding the areas in filling and cutting respectively.

**4. Three-Level Section (Fig. 12.4):**

In this case, the transverse slope of the ground is not uniform.

**5. Multi-Level Section (Fig. 12.5):**

In this case, the transverse slope of the ground is not uniform but-has multiple cross-slopes as is clear from the figure.

**The notes regarding the cross-section are recorded as follows:**

The numerator denotes cutting (+ve) or filling (-ve) at the various points, and the denominator their horizontal distances from the centre line of the~~ ~~section. The area of the section is calculated from these notes by coordinate method. The co-ordinates may be written in the determinant form irrespective of the signs.

Let Σ F= sum of the product of the co-ordinates joined by full lines.

Σ D= sum of the products of the co-ordinates joined by dotted lines.

Then, A= 1/2 (ΣF- ΣD) …………………………………………………….. (Eqn. 12.6)

**Formulae for volume: **

To calculate die volumes of the solids between sections, it must be assumed that they have some geometrical from. They must nearly take the form of prismoids and therefore, in calculation work, they are considered to be prismoids.

Let A_{1}, A_{2}, A_{3}…………….. A_{n} = the areas at the 1st, 2nd, 3rd……………… last cross-section.

D = the common distance between the cross-section.

V = the volume of cutting or filling.

**1. Trapezoidal Formula:**

The number of cross-sections giving the areas may be odd or even. Since the areas at ends are averaged in this formula, therefore, it is also known as Average end Area formula.

**2. Prismoidal Formula: **

In order to apply the prismoidal formula, it is necessary to have odd number of sections giving the areas. If there be even areas, the prismoidal formula may be applied to odd number of areas and the volume between the last two sections may be obtained separately by trapezoidal formula and added.

**Prismoidal Correction: **

The difference between the volumes computed by the trapezoidal formula and the prismoidal formula is known as prismoidal correction. The volume by prismoidal formula is more nearly correct. Since the volume calculated by trapezoidal formula is usually more than that calculated by prismoidal formula, therefore the prismoidal correction is generally subtractive.

Thus volume by prismoidal formula=volume by trapezoidal formula -prismoidal correction.

In the formulae of prismoidal correction given below, the small and capital letters refer to the notations of the adjacent sections. The prismoidal correction is denoted by C_{P}.

**1. ****Level Section:**

**2.Two – Level Section.**

**3. Side- Hill Two –Level Section.**

**4. Three – level Section:**

**Curvature Correction for Volumes:**

The trapezoidal and prismoidal formulae are derived on the assumption that the sections are parallel to each other and normal to the centre line. But when the centre line is on a curve, the~~ ~~sections do not remain parallel to each other and a correction for curvature has to be applied.

This effect is not much pronounced and does not involve large quantities of earth work in ordinary cases, therefore it is neglected. But it has to be considered in final estimates and precise results.

This is quite appreciable in the case of road widening and hill side sections which are partly in cutting and partly in filling. Curved volumes are calculated from Pappu’s theorem. It states that the volume swept by a constant area rotating about a fixed axis is equal to the product of that area and the length of the path traced by the centroid of the area. When the areas are not uniform, mean distance of the centroid from the centre line is taken as

Plus or minus sign indicates that the centroid is on the opposite side or the same side of the centre line as the centre of curvature.

Alternatively, areas are corrected for eccentricity of centroid and the corrected areas are used in prismoidal formula for calculation of volume.

**The curvature corrections (C _{C}) for the common cases are given below:**

**1. Level Section: **

Now correction is necessary in this case since the area is symmetrical about the center axis

**2. Two – Level Section and Three – Level Section:**

**3. Side – Hill Two – Level Section:**

**Measurement of Volumes from Spot Levels: **

This method is used to find excavations in large tracts such as for borrow pits. The field-work consists in dividing up the site of the work into a number of equal triangles, squares or rectangles (Fig. 12.6) and finding the original surface levels and new surface levels after excavation by, spot levelling.

The difference of levels at the original and new surfaces^{ }of a point determines the depth of earth work at that point. The depths of earth work are marked at the corners of the triangles, squares, or rectangles into which the ground is divided.

**The volume of the borrow pit may be obtained by the sum of the volumes of several prisms computed by the following formulae: **

Where A = the horizontal are of the cross-section of the triangular or rectangular prism.

h_{1}, h_{2}, h_{3}, h_{4}, etc. = the depths of excavation marked at the corners.

**Measurement of Volumes from contours: **

**Mass Diagram: **

The mass diagram is a graph plotted between distances along centre line, taken as base and algebraic sum of the mass of the earth work, taken as ordinates. The volume of cutting is considered as positive where as that of filling as negative.

For determining in advance, the proper distribution of excavated material and the amount of waste and borrow, a mass diagram is commonly used. From the mass diagram, it is possible to determine by trial, the earthwork distribution plan that will result in the minimum cost of overhaul and the economical expenditure for overhaul and borrow.

**Lift and Lead: **

**Lift:**

Vertical distance through which the excavated earth is lifted beyond a certain depth is called lift. Excavation up to 1.5m depth below ground level and excavated material deposited on the ground shall be included in the item of work as specified. The lift shall be measured from the C.G. of the excavated earth to that of the deposited earth. Extra lift shall be measured in unit of 1.5m or as per pre-accepted condition.

**Lead:**

The horizontal distance from borrow pit to the site of work is called lead. It shall be measured from the centre of the area of excavation to the centre of the placed earth. Normally a lead upto 30m or as per pre- accepted condition is not paid-extra.

Beyond a lead of 30m and lift of 1.5m rates will be different for every unit of 30m lead and 1.5m lift or fraction thereof.

**Converting Lift into lead:**

**The lift is converted into lead by the following rules: **

1. The lift up to 3 .6m is multiplied by 10

2. Lift more than 3.6m and less than 6m is squared and multiplied by 3.3.

3. Lift more than 6m is multiplied by 20.

**Examples on Earth Work: **

**Example 1:**

**The following are the reduced levels of the consecutive points 30m apart on a longitudinal surface section of a proposed road: **

The formation level at change 0 is lm below the natural surface level and then rises uniformly on a gradient of 1 in 40. Find the corresponding depth of cutting or heights of embankment.

**Solution: **

Since the formation rises on a uniform gradient of 1 in 40, rise in 30m

The formation levels of the successive points may be obtained by adding 0.75m to the formation level of the point preceding it .

**Thus the formation levels at different changes will be as under:**

The differences between the natural surface level and formation level at any point will the depth of cutting or height of embankment at the point.

**Therefore we get: **

**Example 2:**

A railway embankment is 10m wide with side slopes 2:1. Assuming the ground to be level in a direction transverse to the centre line, calculate the volume contained in a length of 150 metres, the central heights at 30m intervals being 2.5, 3.00, 3.5, 4.0, 3.75 and 2.75m respectively.

**Solution:**

Refer to fig .12.2, b= 10m, s=2

The prismoidal formula necessitates odd number of X-sections but in this case they are even. Therefore, volume of the last strip will be found separately by trapezoidal formula which shall be added to the volume of the remaining strips giving odd number of X-sections, found by prismoidal formula to get the total volume.

**Example 3:**

A road embankment 8m wide at formation level with side slopes 2 :1, and an average height of bank 2m constructed with an average gradient of 1 in 30 from a 320m contour to 450m contour, find (i) the length of the road, and (ii) the quantity of earth for embankment.

**Solution:**

**Example 4:**

The width of formation level »f a cer4ain cutting is 10m and the side slopes are 1: 1. The surface of the ground has a uniform side slope of 1 in 6. If the depths of cutting at the centre lines of three~~ ~~section~~s~~ 30m apart are 3m, 4m and 5m respectively, determine the volume of earth work involved in this length of cutting.

**Solution:**

Refer Fig 12.1of the two –level~~ ~~section b=10m; h=3, 4 ,5 m; s=1; r= 6

Volume of prismoid by using trapezoidal formula and applying prismoidal correction = 3529.695 -10.285 = 351941 cum. Same as found above.

**Example 5:**

The width at formation of a certain road is 12m and the side slopes of 1 in 1 in cutting and 1 in 2 in filling. The original ground has a cross-fall of 1 in 5. If the depth of excavation at the centre lines of the two section~~s~~ 50m apart are (1.4 m and 0.8 m respectively, find the volume of cutting and volume of filling over this length.

**Solution: **

Refer Fig 12.3 of side hill two-level section.

**(i) Volume of cutting by trapezoidal formula:**

**(ii) Volume of filling by trapezoldal formula,**