A steel tape not supported along its entire length sags in the form of a catenary, a good example being the cable between two power poles. Because of sag, the horizontal distance (chord length) is less than the graduated distance between tape ends, as illustrated in Figure 6.6. Sag can be reduced by applying greater tension, but not eliminated unless the tape is supported throughout. The following formula is used to compute the sag correction. Where in the English system CS is the correction for sag (difference between length of curved tape and straight line from one support to the next), in feet; LS the unsupported length of the tape, in feet; w the weight of the tape per foot of length, in pounds; and P1 the pull on the tape, in pounds. Metric system units for Equation (6.6) are kg/m for w, kg for P1, and meters for CS and LS.
The effects of errors caused by sag can be eliminated by (a) supporting the tape at short intervals or throughout or (b) by computing a sag correction for each unsupported segment and applying the total to the recorded length according to Equation (6.6). It is important to recognize that Equation (6.6) is nonlinear and thus must be applied to each unsupported section of the tape. It is incorrect to apply it to the overall length of a line unless the line was observed in one section.
As stated previously, when lines of unknown length are being measured, sag corrections are always negative, whereas positive corrections occur if the tension applied exceeds the standard pull. For any given tape, the so-called normal tension needed to offset these two factors can be obtained by setting Equations (6.5) and (6.6) equal to each other and solving for P1.Although applying the normal tension does eliminate the need to make corrections for both pull and sag, it is not commonly used because the required pull is often too great for convenient application.
Tape Not Horizontal and Tape Off-Line
Corrections for errors caused by a tape being inclined in the vertical plane are computed in the same manner as corrections for errors resulting from it being off-line in the horizontal plane. Corrected lengths can be determined by Equation (6.2), where in the vertical plane, d is the difference in elevation between the tape ends, and in the horizontal plane, d is the amount where one end of the tape is off-line. In either case, L is the length of tape involved in the measurement.
Errors caused by the tape not being horizontal are systematic, and always make recorded lengths longer than true lengths. They are reduced by using a hand level to keep elevations of the tape ends equal, or by running differential levels (see Section 5.4) over the taping points, and applying corrections for elevation differences. Errors from the tape being off-line are also systematic, and they too make recorded lengths longer than true lengths. This type of error can be eliminated by careful alignment.
Practice and steady nerves are necessary to hold a plumb bob still long enough to mark a point. The plumb bob will sway, even in calm weather. On very gradual slopes and on smooth surfaces such as pavements, inexperienced tapepersons obtain better results by laying the tape on the ground instead of plumbing. Experienced tapepersons plumb most measurements.
Errors caused by improper plumbing are random, since they may make distances either too long or too short. However, the errors would be systematic when taping directly against or in the direction of a strong wind. Heavier plumb bobs and touching the plumb bob on the ground, or steadying it with one foot, decreases its swing. Practice in plumbing will reduce errors.
Chaining pins should be set perpendicular to the taped line but inclined 45° to the ground. This position permits plumbing to the point where the pin enters the ground without interference from the loop. Brush, stones, and grass or weeds deflect a chaining pin and may increase the effect of incorrect marking. Errors from these sources tend to be random and are kept small by carefully locating a point, then checking it.
When taping on solid surfaces such as pavement or sidewalks, pencil marks or scratches can be used to mark taped segments. Accuracy in taping on the ground can be increased by using tacks in stakes as markers rather than chaining pins.
Incorrect Reading or Interpolation
The process of reading to hundredths on tapes graduated only to tenths, or to thousandths on tapes graduated to hundredths, is called interpolation. Errors from this source are random over the length of a line. They can be reduced by care in reading, employing a magnifying glass, or using a small scale to determine the last figure.
Summary of Effects of Taping Errors
An error of 0.01 ft is significant in many surveying measurements. Table 6.1 lists the nine types of taping errors; classifies them as instrumental (I), natural (N), or personal (P), and systematic (S) or random (R); and gives the departure from normal that produces an error of 0.01 ft in a 100-ft length.
The accepted method of reducing errors on precise work is to make separate measurements of the same line with different tapes, at different times of day, and in opposite directions. An accuracy of 1/10,000 can be obtained by careful attention to details.