Warum ist der Flächenkoeffizient 2α und der Volumenkoeffizient 3α?

Weil die Ausdehnung isotrop (in alle Richtungen gleich) ist. (equal in all directions). Fläche = length × length, so ΔA ≈ 2αA₀ΔT (applying the product rule with small α). Volume = length³, so ΔV ≈ 3αV₀ΔT. These are first-order approximations valid when αΔT ≪ 1, which holds for typical engineering temperature ranges.

Warum werden Dehnungsfugen bei Brücken und Eisenbahnen verwendet?

Stahl dehnt sich bei Temperaturschwankungen stark aus und zieht sich zusammen. A 100 m steel bridge can expand ~33 mm over a 30°C temperature swing. Without expansion joints (gaps designed to accommodate this movement), the resulting thermal stress would buckle rails or crack bridge decks. Expansion joints allow free movement while maintaining structural integrity.

Wie hoch ist der Wärmeausdehnungskoeffizient von Stahl?

Kohlenstoffstahl hat α ≈ 11×10⁻⁶/°C (11 μm/m·°C) und Edelstahl α ≈ 17,3×10⁻⁶/°C. (11 μm/m·°C), and stainless steel has α ≈ 17.3×10⁻⁶/°C. This means a 1-meter steel rod expands by approximately 0.011 mm (11 μm) per 1°C. Over large structures and temperature swings, this small coefficient produces significant absolute expansions that must be engineered for.

Hängt die Wärmeausdehnung von der Anfangstemperatur ab?

Bei den meisten technischen Materialien innerhalb praktischer Temperaturbereiche wird α als ungefähr konstant behandelt., α is treated as approximately constant and the expansion depends only on the temperature change ΔT, not the starting temperature. However, at extreme temperatures (near melting or phase transition points), α changes significantly and the linear approximation breaks down.

Wärmeausdehnungs-Rechner – Linear, Fläche & Volumen

Schnellvorlagen

Material

α (Ausdehnungskoeffizient)

×10⁻⁶/°C

Anfangslänge L₀

Temperaturänderung ΔT

Wählen Sie ein Material aus, geben Sie L₀ und ΔT ein, um die Ausdehnung zu berechnen. Geben Sie für Benutzerdefiniert α direkt ein.

Material

α (×10⁻⁶/°C)

Anfangslänge L₀

ΔT

Select material, enter dimensions and ΔT. For volume of liquids, switch to the Flüssigkeits dropdown.

Was ist Wärmeausdehnung?

Die Wärmeausdehnung ist die Tendenz von Materie, ihr Volumen bei Erwärmung zu vergrößern. Wenn die Temperatur steigt, vibrieren Atome und Moleküle stärker und nehmen mehr Platz ein., pushing each other apart and causing the material to occupy more space. This effect occurs in solids, liquids, and gases.

Bei Festkörpern unterscheiden wir drei Arten: linear (change in one dimension), area (change in two dimensions), and volumetric (change in all three dimensions). Engineering applications must account for thermal expansion in bridges, rails, pipelines, and precision instruments.

ΔL = α·L₀·ΔT

Lineare Ausdehnung

ΔA = 2α·A₀·ΔT

Fläche expansion

ΔV = β·V₀·ΔT

Volumetrisch (β = 3α for solids)

Linear, Fläche, and Volumetrisch Expansion

Die drei Ausdehnungsarten hängen durch die isotrope Natur der meisten Materialien zusammen. of most materials. If a material expands by factor α in every direction:

Typ	Formel	Koeffizient	Beziehung
Linear	ΔL = α·L₀·ΔT	α	—
Fläche	ΔA ≈ 2α·A₀·ΔT	2α	2 × lineares α
Volumetrisch (solid)	ΔV ≈ 3α·V₀·ΔT	β = 3α	3 × lineares α
Volumetrisch (liquid)	ΔV = β·V₀·ΔT	β (direct)	Direkt gemessen

Koeffizient Table for Common Materials

Material	α (×10⁻⁶/°C)	β = 3α (×10⁻⁶/°C)
Aluminium	23.1	69.3
Stahl (Kohlenstoff)	11.0	33.0
Beton	12.0	36.0
Kupfer	17.0	51.0
Glas (Pyrex)	3.3	9.9
Glass (window)	8.5	25.5
Titan	8.6	25.8
Wolfram	4.5	13.5
Quarz (geschmolzen)	0.59	1.77
Wasser (liquid)	—	210 (β direct)
Ethanol	—	1100 (β direct)

Anomalous Expansion of Wasser

Wasser behaves unusually between 0°C and 4°C — it contracts as temperature rises (negative β in this range), reaching maximum density at exactly 4°C. Below 4°C, it expands as it cools, and below 0°C, ice is about 9% less dense than liquid water.

This anomaly is crucial for aquatic life: lakes freeze from the top down (ice floats), preserving liquid water beneath. The thermal expansion calculator's water preset uses β ≈ 210×10⁻⁶/°C (valid above 4°C at ~20°C reference).

Technische Anwendungen (Brücken, Schienen, Rohre)

Brücken: Steel bridge girders can expand tens of millimetres across a seasonal temperature range of 50°C+. Expansion joints (roller bearings and sliding plates) allow free movement without buckling.

Eisenbahnen: Continuous welded rail (CWR) is pre-stressed during installation at a neutral temperature so that rail stress remains compressive in summer and tensile in winter, avoiding buckling and cracking.

Rohrleitungen: Expansion loops and bellows compensate for thermal growth in steam lines and oil pipelines that operate at high temperatures relative to their installation conditions.

Bimetallstreifen: Two metals with different α are bonded together. When heated, the strip bends toward the metal with the lower α. Used in thermostats and circuit breakers.

Berechnungsbeispiele

Beispiel 1 — Stahlbrücke, 100m, +30°C

α = 11×10⁻⁶/°C, L₀ = 100 m, ΔT = 30°C

ΔL = 11e-6 × 100 × 30

ΔL = 0.033 m = 33 mm

Example 2 — Aluminium rod, 1m, +100°C

α = 23.1×10⁻⁶/°C, L₀ = 1 m, ΔT = 100°C

ΔL = 23.1e-6 × 1 × 100

ΔL = 0.00231 m = 2.31 mm

Example 3 — Beton area, 10m², +40°C

α = 12×10⁻⁶/°C, A₀ = 10 m², ΔT = 40°C

ΔA = 2×12e-6×10×40

ΔA = 9.6×10⁻³ m² = 96 cm²

Example 4 — Wasser tank, 1000L, +50°C

β = 210×10⁻⁶/°C, V₀ = 1000 L, ΔT = 50°C

ΔV = 210e-6 × 1000 × 50

ΔV = 10.5 L (1.05% volume increase)

Häufig gestellte Fragen

Die Wärmeausdehnung ist die Tendenz von Materie, ihre Form, Fläche und ihr Volumen als Reaktion auf eine Temperaturänderung zu ändern., area, and volume in response to a change in temperature. When a material is heated, its particles move faster and occupy more space, causing the material to expand. The amount of expansion depends on the material's coefficient (α or β) and the temperature change ΔT.

Der lineare Wärmeausdehnungskoeffizient (α) beschreibt, wie stark sich die Länge eines Materials pro Längeneinheit pro Grad Temperaturänderung ändert. (α) describes how much a material's length changes per unit length per degree of temperature change. It is expressed in units of 1/°C (numerically the same as 1/K). For example, steel has α ≈ 11×10⁻⁶/°C, meaning a 1-meter rod expands by 11 micrometers for every 1°C rise.

Weil die Ausdehnung isotrop (in alle Richtungen gleich) ist. (equal in all directions). Fläche = length × length, so ΔA ≈ 2αA₀ΔT (product rule, keeping only first-order terms). Volume = length³, so ΔV ≈ 3αV₀ΔT. These approximations are accurate when αΔT ≪ 1, which holds for all practical engineering temperatures.

Im Gegensatz zu den meisten Flüssigkeiten dehnt sich Wasser aus, wenn es unter 4 °C abgekühlt wird. This is because water molecules form a hexagonal hydrogen-bond lattice when freezing, which is less dense than liquid water. This anomaly allows ice to float, keeping lakes from freezing solid and enabling aquatic life to survive winter.

Steel expands significantly with temperature. A 100 m bridge can expand ~33 mm over a 30°C swing. Without expansion joints, the resulting thermal stress would buckle rails or crack bridge decks. Expansion joints allow free movement while maintaining structural integrity and safety.

Carbon steel has α ≈ 11×10⁻⁶/°C and stainless steel has α ≈ 17.3×10⁻⁶/°C. This means a 1-meter carbon steel rod expands by approximately 0.011 mm per 1°C. Over large structures and wide temperature ranges, this produces centimetres of total expansion that engineers must account for.

For most engineering materials in practical temperature ranges, α is treated as approximately constant and expansion depends only on ΔT, not the starting temperature. However, near melting or phase-transition points, α changes significantly and the linear approximation breaks down.

α (Alpha) ist der lineare Wärmeausdehnungskoeffizient: ΔL = α·L₀·ΔT.: ΔL = α·L₀·ΔT. β (beta) is the volumetric thermal expansion coefficient: ΔV = β·V₀·ΔT. For isotropic solids, β = 3α. For liquids, β is measured directly since liquids have no fixed shape. Wasser has β ≈ 210×10⁻⁶/°C at 20°C, while ethanol has β ≈ 1100×10⁻⁶/°C.