Calculation
ΔL = ρ · g · L² / (2E)
At the top the rod carries the weight of everything below; at the free bottom the axial force is zero. Integrating strain gives an extension proportional to length squared.
Integrierte Eigengewichtsdehnung · ΔL = ρgL²/(2E)
Calculate Hanging Rod Extension from Self-Weight
At the top the rod carries the weight of everything below; at the free bottom the axial force is zero. Integrating strain gives an extension proportional to length squared.
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Result
Select a target and calculate.
Understand the inputs
- Extension ΔL — Increase in length of the freely hanging rod caused by its own weight alone. Include external loads and temperature change separately for actual end position.
- Material density ρ — Mass per volume of the rod material from a datasheet; steel is roughly 7850 kg/m³ as a guide.
- Gravitational acceleration g — Local gravitational acceleration; near 9.81 m/s² on Earth.
- Free rod length L — Vertical length from the top support to the unloaded free bottom. Long rods or cables are particularly affected because L enters squared.
- Young modulus E — Stiffness of the homogeneous material at operating temperature from a datasheet; steel about 210 GPa.
Example
A 100 m steel rod with ρ = 7850 kg/m³ and E = 210 GPa extends under g = 9.80665 m/s² by ΔL = ρgL²/(2E) = 1.833 mm.
Assumptions and limits
Straight homogeneous vertical constant-section rod fixed at top, free and unloaded at bottom, with linear elastic strain at constant density and E. Temperature, creep, cable slip, external loads and varying section are excluded.
Understand Extension of a hanging rod under its own weight
A long freely hanging rod stretches under its own weight. Its top carries more weight than its bottom, so strain is not constant along its length.
What does this quantity describe?
At each cross-section the rod carries the portion below. Axial tension rises linearly from zero at the free bottom to total weight at the top. Local strain equals axial stress divided by Young modulus E. For homogeneous density ρ, constant section and length L, integration gives ΔL = ρgL²/(2E). Here ΔL is extension and g is gravitational acceleration. Section area cancels because weight and axial stiffness both grow with it.
Formula and variables
ΔL = ρ · g · L² / (2E)
Axial force at height x from free bottom: N(x) = ρgAxLocal strain: ε(x) = ρgx/ETotal extension: ΔL = ρgL²/(2E)
| Symbol / input | Meaning |
|---|---|
| Extension ΔL | Increase in length of the freely hanging rod caused by its own weight alone. Include external loads and temperature change separately for actual end position. |
| Material density ρ | Mass per volume of the rod material from a datasheet; steel is roughly 7850 kg/m³ as a guide. |
| Gravitational acceleration g | Local gravitational acceleration; near 9.81 m/s² on Earth. |
| Free rod length L | Vertical length from the top support to the unloaded free bottom. Long rods or cables are particularly affected because L enters squared. |
| Young modulus E | Stiffness of the homogeneous material at operating temperature from a datasheet; steel about 210 GPa. |
Choose the inputs correctly
Get ρ in kg/m³ and E in pascals from material data for the operating condition; for steel 7850 kg/m³ and 210 GPa are only rough guides. L is free vertical length from suspension to the unloaded bottom. On Earth g is about 9.81 m/s². The formula requires constant section area.
How to use the calculator
Enter density, modulus, free length and local g. Add calculated extension to unloaded length. If an extra end force acts, calculate its extension separately and superpose only within the linear range.
Worked example
For a 100 m steel rod with ρ = 7850 kg/m³, E = 210 GPa and g = 9.80665 m/s², ΔL = 7850·9.80665·100²/(2·210·10⁹) = 0.001833 m = 1.833 mm. At 200 m the extension is four times as large.
Understand the result and units
Self-weight extension grows with L squared but does not depend on constant section area. A thicker homogeneous rod weighs more but gains axial stiffness in the same proportion.
The internal calculation uses kg/m³, metres, pascals and m/s². Display can use mm or other offered length units.
Typical applications
Plausibility checks for long hanging rods, measuring members and tension members before including other loads.
Assumptions, limits and common mistakes
Straight vertical homogeneous rod of constant section, fixed at top and free and unloaded at bottom only. Linear elasticity, constant temperature and small strains are assumed. Attached loads, creep, cable construction, motion and lateral loads are excluded.
Common mistake: Do not substitute total weight into ΔL = FL/(EA): that doubles self-weight extension because axial force vanishes at the bottom. L is free vertical length and E should represent operating temperature.
Frequently asked questions
What is “Extension of a hanging rod under its own weight” used for?
Plausibility checks for long hanging rods, measuring members and tension members before including other loads.
Where do the input values come from?
Get ρ in kg/m³ and E in pascals from material data for the operating condition; for steel 7850 kg/m³ and 210 GPa are only rough guides. L is free vertical length from suspension to the unloaded bottom. On Earth g is about 9.81 m/s². The formula requires constant section area.
What does the result not cover?
Straight vertical homogeneous rod of constant section, fixed at top and free and unloaded at bottom only. Linear elasticity, constant temperature and small strains are assumed. Attached loads, creep, cable construction, motion and lateral loads are excluded.