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# Answer to Question #155309 in Civil and Environmental Engineering for Ian

Question #155309

A line is recorded as 475.25 m long. It is measured with a 6.5 N tape which is 30.492 m long at 680F under a 10 lb. pull supported at end points. During measurement the temperature is 4.50C and the tape is suspended under a 71 N pull. The line is measured on 3% grade. What is the true horizontal distance? Modulus of elasticity of the tape is 1.93 X 10KPa and cross-sectional area of tape is 0.0284 cm2. Coefficient of expansion of the tape material is 0.0000116 m/0C.

1
2021-01-20T04:56:31-0500

Distance Measured on the ground, Dm=475.25 m

Grade on which line is measured = 3%

Modulus of Elasticity of tape, E= 200 GPa =2*1011 N/m2

Coefficient of Linear Expansion, "\\alpha" = 0.0000116 m/oC

C/S Area of the tape, A= 0.0284 cm2 =2.84*10-6 m2

Standard Length, Lo=30.492 m

Standard Temperature, To=68oF=20oC

Standard Pull, Po= 10 lb = 44.5 N

During measurement

Weight of the Tape, W=6.5 N.

Temperature, Tm=4.5oC

Pull, Pm=71 N

Correction due to temperature is given by; "C_{temp}=L_o \\alpha (T_m-T_o)"

"C_{temp}=30.492\\times0.0000116(4.5-20)= -0.0054824616m"

Correction due to Pull is given by; "C_{pull}=\\frac{(P_m-P_o)L_o}{A.E}"

"C_{pull}=\\frac{(71-44.5)\\times 30.492}{2.84\\times10^{-6} \\times2\\times10^{11}}=0.00142260211m"

Correction due to Sag is given by"C_{sag}=-\\frac{W^2L_0}{24P_m^2}"

"C_{sag}=-\\frac{6.5^2 \\times 30.492}{24 \\times 71^2}=-0.01064840805 m"

Net Correction applied on the tape,"C_{net}=C_{temp}+C_{pull}+C_{sag}"

"C_{net}=-0.0054824616m+0.00142260211+0.01064840805= -0.014708268\n m"

Hence, the Actual Length of the tape is given by:"L=L_o+C_{net}=30.492 -0.014708268=30.47729173 m"

Total Error to be corrected for inclined distance:"E_{slope}=D_t\\times(1-cos\\theta)"

But "D_t=\\frac{30.492}{30.47729173}\\times475.25=475.47835454303"

"E_{slope}=475.47835454303\\times(1-cos(tan^{-1}0.03))=+0.21382094123 m"

True Horizontal Distance: "D=D_t+C{slope}"

"D_t+E{slope}=475.47835454303+0.21382094123=475.6921755m"

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