SME
Surveying · Problem type
Temperature and sag corrections of a steel tape
Use this for “A steel tape reads 29.876 m between two pegs at 38 °C, hanging free under 100 N; it weighs 0.2 N/m. Find the true length.”
This is an example. Type your own values to start a new problem.
Cₜ (mm)
=
11.6
×
10⁻⁶
×
L (m)
×
(
T (°C)
−
Tₛ (°C)
)
Cₛ (mm)
=
−
(
w (N/m)
)
2
×
(
L (m)
)
3
÷
(24
×
(
P (N)
)
2
)
Lc (m)
=
L (m)
+
Cₜ (mm)
+
(
Cₛ (mm)
)
Formulas
Cₜ
=
11.6 ×
10
-6
×
L
×
(
T
−
Tₛ)
+0.006238
=
11.6 ×
10
-6
×
29.876
×
(38
−
20)
Cₛ
=
−
w
2
×
L
3
24 ×
P
2
−0.004444
=
−
0.2
2
×
29.876
3
24 ×
100
2
Lc
=
L
+
Cₜ
+
Cₛ
29.8778 = 29.876 + 0.006238 + (−0.004444)
Formulas are worked in m. The step-by-step shows the conversions.
Assumptions
•
A steel tape (11.6 × 10⁻⁶ per °C) standardized at Tₛ, lying fully supported.
•
The span measured hangs free between its two ends, in one piece.
•
A warm tape grows and reads short (Cₜ +); a sagging tape reads long (Cₛ −).
•
Pulled at its standard tension, so there is no tension correction.
Step-by-step
WE KNOW
L = 29.876 m, T = 38 °C, Tₛ = 20 °C, w = 0.2 N/m, P = 100 N
FIND
temperature correction (Cₜ), sag correction (Cₛ), corrected length (Lc)
Step 1 · Find temperature correction (Cₜ)
Use Cₜ = 11.6 × 10⁻⁶ × L × (T − Tₛ)
Steel grows 11.6 millionths of its length for each degree above standard.
Cₜ
=
11.6 ×
10
-6
×
29.876
×
(38
−
20)
Cₜ
=
(11.6
×
10
-6
)
×
29.876
×
18
Cₜ = 0.0003466 × 18
Cₜ = 0.006238 m
Step 2 · Find sag correction (Cₛ)
Use Cₛ = −w² × L³ ÷ (24 × P²)
The hanging tape bows, so it spans less than it reads.
Cₛ
=
−
0.2
2
×
29.876
3
24 ×
100
2
Cₛ
=
−0.04
×
26,666.5819
240,000
Cₛ
=
−
1,066.6633
240,000
Cₛ = −0.004444 m
Step 3 · Find corrected length (Lc)
Use Lc = L + Cₜ + Cₛ
Add both corrections, each with its sign, to the reading.
Lc = 29.876 + 0.006238 + (−0.004444)
Lc = 29.8778 m
Step 4 · Convert the answers
Change each answer back to the unit you chose.
Cₜ = 0.006238 m = +6.2381 mm (1 m = 1000 mm)
Cₛ = −0.004444 m = −4.4444 mm (1 m = 1000 mm)
Check
Put the numbers back in. Both sides should match.
+0.006238
=
11.6 ×
10
-6
×
29.876
×
(38
−
20)
✓
−0.004444
=
−
0.2
2
×
29.876
3
24 ×
100
2
✓
29.8778 = 29.876 + 0.006238 + (−0.004444) ✓
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