Considering theentire area By offsets to the baes line Based on field measurements By latitudes &departures Area Based on measurements scaled from a map By coordinates Instrumental method Usually by a planimeter Based on field measurements – considering the entire area In this method, the area is divided into number of geometrical figures such as triangles, rectangles, squares and trapeziums and then the area can be calculated according to one of following method. Computing the area can be achieved in two steps. 1 st step Area of a triangle: Area= √ s ( s−a ) ( s−b )( s−c ) Where a, b and c are sides and s = (a + b + c) / 2 Area of rectangle Area=a ×b Where a and b are sides. Area of the trapezium 1 Area= ( a+b ) ×d 2 Where a and b are the parallel sides and d is the perpendicular distance between them. 02=ordinates x 1 .Area of square Area=a2 Where a is the side of the square. X1 X22 01 . x 2=chainages Area of the shaded portion= 01+ 02 × ( x 2−x 1 ) 2 ∴Total area=area of the geomatrical shape +area of the boundary . 2 n d step The area along the boundaries is calculated as follows. of ordinates *length of base line Area=o1+o2+…+on/o (n+1)*l Trapezoidal rule This rule is based on the assumption that the figures are trapeziums. The base line is divided into a number of divisions and the ordinates are measured at the mid points of each division. we assume that the boundaries between the extremities of the ordinates are straight lines. Total area= o0+o1/2*d + … . Area=sum of ordinate/no. The offsets are measured from the boundary to the base line or a survey line at regular intervals. There are 4 methods in order to calculate the area. The offsets are measured to each of the points of the divisions of the base line. Mid-ordinate rule Average-ordinate rule Trapezoidal rule Simpson’s rule Mid-ordinate rule In order to apply this method. This rule is more accurate than the previous two rules which are approximate versions of this rule. This rule can be applied for any number of ordinates.Areas from offsets to a baseline – offsets at regular intervals This method is suitable for long narrow strips of land. Area enclosed by one trapezium=o0+o1/2*d Total area is the sum of each separate trapezium. This method can also be applied to a plotted plan from which the offsets to a line can be scaled off. Area=common distance*sum of mid ordinates Mid ordinate=o1+o2/2=h1 Area=d (h1+h2+…) Average-ordinate rule In this method also we assume that the boundaries between the extremities of the ordinates are straight lines. 1st method In this method. Areas from offsets to a baseline – offsets at irregular intervals There are two methods to find the area when the ordinates distances are irregular. Even though this method gives more accurate results out of other three methods. Simpsons’ rule In this rule. this is only applicable when the number of divisions is even. 2nd method Area by coordinates Latitudes and departures – double meridian distances This method is one of the most frequently used for computing the area of a closed traverse. This method is more useful when the boundary line departs considerably from the straight line. Multiply the total sum thus obtained by the common distance between the ordinates to get the required area. the whole multiplied by one-third the common interval between them. the area of each trapezoid is calculated separately and then added together to compute the total area. we assume that the short lengths of boundary between the ordinates are parabolic arcs.Total area=d/2*(o1+2o2+…+2on-1+on) Total area=common distance/2*(1st ordinate + last ordinate+2[sum of other ordinate] Add the average of the end offsets to the sum of the intermediate offsets. . The area is equal to the sum of the two end ordinates plus four times the sum of the even intermediate ordinates plus twice the sum of the odd intermediate ordinates. The meridian distance of any line is equal to the meridian distance of the preceding line plus half the departure of the preceding line plus half the departure of the line itself. method are identical.M. Area of triangle or trapezium=latitude of the line*meridian distance of the line Double meridian distance The double meridian distance of a line is equal to the sum of the meridian distances of the two lines. Area by latitudes and meridian distances East-west lines drawn from each station to the reference meridian. of the preceding line plus the departure of the preceding line plus the departure of the line itself. . The D. Meridian distances The meridian distance of any point in a traverse is the distance of that point to the reference meridian.M. the base of the triangle or trapezium will be the latitude of the line. Area by latitudes and double meridian distances A=area of dDcC+area of CcbB-area of dDA-area of ABb Area from departures and total latitudes Area by double parallel distances and departures A parallel distance of any line of a traverse is the perpendicular distance from the middle point of that line to a reference line (chosen to pass through most southerly station) at right angles to the meridian. the latitudes and departures of each line of the traverse are calculated.D method & D.D.P.In order to calculate the area by this method. and the height of the triangle or trapezium will be the meridian distance of that line. The principle of finding area by D. The DPD of any line is the sum of the parallel distances of its ends. A reference meridian is then assumed to pass through the most westerly station of the traverse & the double meridian distances of the lines are computed. Area by coordinates This method can be applied when the offset intervals are irregular. thus getting triangles and trapeziums. of any line is equal to the D. One side of each triangle or trapezium will be one of the lines.D. measured at right angles to the meridian. Then the traverse is balanced.M.D. By observing the disc. The area obtained from this instrument is more accurate than other graphical methods. wheel and Vernier the initial reading (IR) is recorded. Finally. It should be ensured that the tracing point is easily able to reach every point on the boundary line. A good starting point is marked on the boundary line. observing the disc. Instrumental method The instrument used for computation of area from a plotted map is the planimeter. The tracing point is moved gently in a clockwise direction along the boundary of the area. the area of the figure may be obtained from the following expression.The procedure for this method is that from the given distances & offsets. Then the coordinated of all other points are arranged with reference to the origin. Then. The two types of planimeter are: Amsler polar planimeter Roller planimeter Procedure of finding the area with a planimeter The Vernier of the index mark is set to the exact graduation marked on the tracer arm corresponding to the scale as obtained from the table. The anchor point is fixed firmly in the paper outside or inside the figure. The number of times the zero mark of the dial passes the index mark in a clockwise or anticlockwise direction should be observed. wheel and Vernier the final reading (FR) is recorded. Are=M (FR-IR+.10N +C) M=multiplier given in the table N=number of times the 0 mark of the dial passes the index mark C=constant given in the table . a point is selected as the origin. Level section Two-level section From cross sections Side hill two-level section Three level section Volume From spot levels Multi-level section By cross sections From contours By equal depth contours By horizontal planes Volume calculation from cross sections This is the most widely used method. By measuring graphically from plotted cross-sections drawn to scale. In order to compute the volume it is first necessary to evaluate the cross-sectional areas. Cross-sections are established at some convenient intervals along a center line of the works. areas being obtained by planimeter or division into triangles or square. Volumes are calculated by relating the cross-sectional areas to the distances between them. which may be obtained by the following methods: By calculating from the formula or from first principles the standard cross-sections of constant formation widths and side slopes. Level section Two level sections Side hill two level sections . The total volume is divided into a series of solids by the planes of cross sections. The various cross sections may be categorized as. Depth of centre line or height of embankment = h Formation width = b Side width = w Area = h(b + mh) Two level sections .level section Level section In this case the ground is level transversely.Three level sections Multi. the ground slope crosses the formation level so that one portion of the area is in cutting and the other in filling. Area of fill = ½ [(b/2 + kh)2/(k-m)] Area of cut = ½ [(b/2 .Area = 1/2m [(b/2 + mh)(w1 + w2) – b2/2] Side hill two level sections In this case.kh)2/(k-n)] . The prismoidal formula A prismoid is defined as a solid whose end faces lie in parallel and consist of any two polygons. parallelograms or trapezium. The longitudinal faces take the form of triangles. . the longitudinal faces being surface extended between the end planes. The total volume of the pyramid can be stated as follows. Where h is the length of the prismoid measured perpendicular to the two end parallel planes and. not necessarily of the same number of sides.Three level sections Area = ½ m [(w1+w2) (mh+b/2) – b2/2) The volumes of the prismoids between successive cross-sections are obtained either by trapezoidal formula or by prismoidal formula. v = h [ ( A1 + A2) / 2 ] but this is true only if the prismoid is composed of prisms and wedged only and not of pyramids. If there are even numbers of sections. a2. The trapezoidal formula (Average End Area Method) It is no more accurate to use the prismoidal formula where the mid-sectional areas have not been directly measured than it is to use the end areas formula. Hence this method is based on the assumption that the mid-area is the mean of the end areas. …. In some cases the volume is calculated and then a correction is applied the correcton being equal to the difference between the volume as calculated and that which could be obtained by the use of the prismoidal formula. In that case the volume of the above prismoid is given as. an spaced at a constant distance h apart. Total voume=h/3[(a1+an)+4(a2+a4…an-1)+2(a3+a5…an-2)] This is also known as Simpson’s rule for volume. particularly as the earth solid is not exactly represented by a prismoid. it is necessary to have an odd number of cross-sections. the correction is known as the prismoidal correction.A1=area of cross-section of one end plane. the end strip must be calculated separately. In order to apply this formula. and the volume between the remaining sections may be calculated by prismoidal formula. A=the mid area In order to calculate the volume of earth work between a number of sections having area a1. a3. . A2= area of cross-section of other end plane. an spaced at a constant distance h apart. a3. Therefore for a rectangle with corner depth ha. …. rectangles or triangles and measuring the levels of their corners before and after the construction. V = Area of the figure × average depth. the depth of excavation or height of filling at every corner is known. a2. Thus.In order to calculate the volume of earth work between a number of sections having area a1. . hb. V = d{[( A1 + A2) / 2 ] + A2 + A3 + …… An-1} level section prismoidal correct Cp= two sections level side hill two three level sections sections level D×s h1−h2 )2 ( 6 ion Volume from spot levels In this method. hc and hd. the field work consists in dividing the area into a number fo squares. the assumption that there is an even slope between the contours is incorrect and volume calculation from contours become unreliable. The area of the cut and fill can be found from the cross-section. The volumes of earth work between adjacent cross-sections may be calculated by the use of average end areas. Unless the contour interval is less than 1m or 2m at the most. There are four distinct methods depending upon the type of work. However. . A=horizontal area of the cross-section of the prism Volume of a group of triangles having the same area Volume from contour plan Contour lines may be used for volume calculations and theoretically this is the most accurate method. high accuracy is not often obtained.Volume of a group of rectangles or squares having the same area Σh1 = some of depths used once Σh2 = sum of depths used twice Σh3 = sum of depths used thrice Σh4 = sum of depths used four times. as the small contour interval necessary for accurate work is seldom provided due to cost. By cross-sections On the same cross-section used to draw the ground surface the grade line of the proposed work can be drawn and the area of the section can be estimated either by ordinary methods or with the help of a planimeter. where the contours of the finished surface intersect a contour of the existing surface the cut or fill can be found by simply subtracting the difference elevation between the two contours. By joining the points of equal cut or fill. H=contour interval V-total volume V=sigmah/2 (a1+A2) by trapezoidal V=Sigma h/3(A1+4A2+A3) by prismoidal formula By horizontal planes This method consists in determining the volumes of earth to be moved between the horizontal planes marked by successive contours. . the contours of the finished or graded surface are drawn on the contour map at the same interval as that of contours.By equal depth contours In this method. These lines are the horizontal projections of the lines cut from the existing surface by planes parallel to the finished surface. At every point. a set of lines is obtained. The volume between any two successive areas is determined by multiplying the average of the two areas by the depth between them. or by the prismoidal formula. 4 9.23 6 4.81 9.0 8.0 20.5 8.34 6.65 4.89 7.Chainag e along the survey line Ordinate (m) 0.0 100.81 6.5 7. .0 80.5 4 2 0 0 20 40 60 80 100 120 140 160 180 200 In this particular survey line there are two distinct Chainage intervals of 20m and 30m.89 7.3 6.0 160.0 60.5 According to the given data above we can plot a graph to determine the enclosed area of the given survey line.23 7.0 130. Therefore when calculating the area it has to be separated into two parts.0 190.0 40.34 7. The method used to calculate the area is the trapezoidal rule since it is more accurate than the mid-ordinate rule and average-ordinate rule and Simpson’s rule cannot be applied because the number of divisions is not even in the plotted plan. 12 10 9.63 6.4 8 7.3 9. Y-Values 12 10 8 Axis Title Y-Values 6 4 2 0 0 20 40 60 80 Axis Title 100 130 160 190 .
Report "area and earthwork volume calculation methods for surveyors"