<< Previous Section Table of Contents The Polar Planimeter Page Numbers Next Section >>


Quantities of Materials.

I. Volumes from Cross Sections.

3. Volumes of Continuous Prismoids.

b. Volumes from Field Notes direct.

Method of Measurement and Use of the Template.

It is evident that when the template is adjusted to the Cross section sheet in the manner described, the tracing of each cross-section with the Planimeter is easily accomplished from the mere inspection of the field notes of each section: the width of road-bed and slope ratios of all the sections in any given case being the same it is a very easy matter with the template to locate on the cross-section sheet both the points of intersection of each side slope with the surface of the ground and the center height and to conduct the tracing needle from one to another of the points thus located, the edges of the template causing the needle to accurately trace the other three sides of the section. For example, suppose we wish to trace the Cross-section at Station 0 in the manner described, the operation would be as follows: Having selected a point of beginning say the point of intersection of road-bed and the left hand slope, we adjust the planimeter to its most favorable position and place the tracer at the point of beginning. We then begin the tracing by moving the tracer along the edge of the template up the left hand slope of the section until we have arrived at the point on the cross-section sheet indicated by the expression 34.0 ÷ 12.0.

The mental location of this point, which is the point of intersection of the given slope with the surface of the ground, is made with the greatest ease and rapidity by simply determining with the eye the horizontal line at the numbered center line which represents a height of 12 ft. above the Road bed and following it until it intersects the edge of the template at the side slope.

The needle being brought to this point, it is then conducted along an undrawn straight line to that point on the center line which is 16.0 above the road bed. From this it is conducted in a straight line to that point on the right hand slope, where the horizontal line 14 ft. above the road bed intersects the right hand slope, which point is indicated by the expression 37.0 ÷ 14.0; and from this point the tracer is simply carried along the edge of the template to the point of beginning, thus tracing the outline of the given cross section.

It will take but little experience to enable the three surface points of each section to be accurately located mentally and with a rapidity which should cause no stop or hesitation in the tracing and the measurement of volumes and quantities by this method, and with the use of the template will be found to be quite as accurate as is attainable with the plotted cross sections and without the time and mental effort expended in plotting the Cross-sections necessitated by the first method.

As already stated, the operations involved in both plotted section and template measurements are the same and have already been explained in detail; they consist in adjusting the Planimeter to the Setting given in the tables for the scale of the Cross-section paper used and then tracing once each the first and last cross-sections of the set of prismoids which are to be included in the one measurement, dividing the Reading due to the tracing of these two end sections by two and setting the resulting value down as a partial result. Then, as previously explained, having readjusted the Planimeter to a Zero Reading each even numbered section is traced once and each odd numbered section traced twice, the entire operation being continuous and without intermediate Reading; and after subtracting from the total Reading the value first set aside, the result is multiplied by the tabular value of the relative vernier unit for the scale used, and the product thus obtained is the total volume of the prismoids measured.

In rock work, or where a greater degree of accuracy is desired in obtaining the volumes of materials, it is often customary to take the field notes of the Cross-sections in greater detail by including in the cross-section notes the elevation of the surface of the ground above the Road bed at a distance out on each side of the Center line equal to one-half the width of the road bed, thus making the cross-sections “five level sections,” instead of “three level sections,” as used in the above example.

The location of these two points is quite as accurately and easily found on the cross-section sheet as are the other three, and admit of “five level sections” being traced quite as readily as are the “three level.”

It may be urged, and especially in the case of large sections, that in tracing the surface of the ground of each section, since no lines are drawn, the tracer may deviate more or less from a straight path between the slope stake and the Center or the Center and the other slope stake, but it is readily seen that the actual surface of the ground is rarely a straight line, and that the deviating path of the tracer would probably conform to the actual surface with quite as much accuracy as though following a drawn straight line connecting the points.

<< Previous Section Table of Contents The Polar Planimeter Page Numbers Next Section >>