Tiêu đề 108 Tape Corrections.pdf ✅

MSTC 108: Tape Corrections 1. Change in Length Consider a single standard tape with supposed (or “true”) length l. If it is shorter by a bit, by a length C, then the reading (or “nominal length”) will be NL = l + C. l = NL − C Similarly, if a tape is too long by a bit by a length C, then the reading will be NL = l − C. l = NL + C Most of the time, a course is not measured exactly by only one whole length of a standard tape. Thus, if the tape is measured by times NL l (for example, a 200 m line is measured by a 50 m tape 200 ÷ 50 = 4 times), then there are also NL l times the correction. TL = NL + C ( NL l ) (tape too long) TL = NL − C ( NL l ) (tape too short) 2. Correction due to Temperature From MSTC 99: Thermodynamics, the expansion of an object due to a temperature change is ∆L = αL∆T If a tape is manufactured to be length l at a standard temperature Ts , then the correction due to temperature change at a temperature T is given by Ct = αL(T − Ts ) The tape is hotter than normal and expands if T is higher than Ts . The correction is also positive in this case. The tape is colder than normal and contracts if T is lower than Ts . The correction is also negative in this case. For the true length, TL = NL + Ct ( NL l ) Note that the correction is always added to the nominal length since its sign already adjusts itself based on the case (positive for tape too long, negative for tape too short).
3. Correction due to Pull From PSAD 9: Axial Deformation, the lengthening of an object due to extensive pull is ∆L = PL AE If a tape is manufactured to be length l at a standard pull Ps , then the correction due to tension change at a pull P is given by CP = (P − Ps )L AE The tape is pulled too hard and stretches if P is higher than Ps . The correction is also positive in this case. The tape is pulled too light and sags if P is lower than Ps . The correction is also negative in this case. For the true length, TL = NL + CP ( NL l ) Note that the correction is always added to the nominal length since its sign already adjusts itself based on the case (positive for tape too long, negative for tape too short). 4. Correction due to Sag The shape of any freely bending object carrying only its weight can be characterized by a catenary (see PSAD 5: Analysis of Structures) with an equation. y = a cosh x a where a = P w . For the length of the tape (see MSTC 78: Length of Arc), y ′ = sinh x a l = ∫ √1 + ( dy dx) 2 dx NL 2 − NL 2 = ∫ √1 + sinh2 x a dx NL 2 − NL 2 l = 2a sinh NL 2a Since sagging shortens the tape, l = NL − C C = NL − 2a sinh NL 2a Approximating the value of the correction using power series, C = − (NL) 3 24a 2
Simplifying using a = P w and NL = L, Cs = − w2L 3 24P2 Where: w = lineal weight of the tape in N/m L = unsupported length P = pull on the tape in N Considering the whole weight of the tape, W = wL, Cs = − W2L 24P2 If the line is measured in multiple segments, total correction = −∑ w2L 3 24P2 + (− w2L2 3 24P2 ) For the horizontal distance, TL = NL + Cs The correction due to sag is added even if it shortens the tape because the correction is already negative. 4.1. Normal Pull Since sagging always shortens the tape, an extensive pull can be applied to stretch the tape such that the corrections cancel each other out and the observed length equals the standard. The pull that cancels the effects of sagging is called normal pull. CP + Cs = 0 (PN − Ps )L AE − w 2L 3 24PN 2 = 0 (PN − Ps )L AE = w2L 3 24PN 2 5. Correction due to Slope When measuring distances between stations, the horizontal distance is the one being required. Along slopes, the measured distance is a bit longer than the horizontal distance. Ch = S(1 − cos θ)