Torsional Constant Calculator
Calculate torsional constants for various cross-sectional shapes and structural profiles
Calculate Torsional Constant
K = ab³/3 - 0.21b⁴ - 0.0175b⁸/a⁴
Torsional Constant Results
Cross-Section: Solid Rectangle
Formula: K = ab³/3 - 0.21b⁴ - 0.0175b⁸/a⁴
Application: Used in torsional stiffness calculations and angle of twist analysis
Engineering Notes
Example Calculation
Rectangular Cross-Section Example
Cross-section: Solid rectangle
Width (a): 10 mm
Height (b): 5 mm
Formula: K = ab³/3 - 0.21b⁴ - 0.0175b⁸/a⁴
Calculation
K = (10)(5³)/3 - 0.21(5⁴) - 0.0175(5⁸)/(10⁴)
K = (10)(125)/3 - 0.21(625) - 0.0175(390,625)/10,000
K = 416.67 - 131.25 - 0.68
K = 284.74 mm⁴
Cross-Section Guide
Circle
Most efficient for torsion, no warping
Rectangle
Common structural shape, warping occurs
I-Beam
Excellent for bending, limited torsional strength
Thin-Walled
Efficient material use, special analysis required
Torsional Design Tips
Circular sections are most efficient for pure torsion
Consider warping restraint in end conditions
Thin-walled sections can buckle under torsion
Larger dimensions increase torsional stiffness dramatically
Use angle of twist to check serviceability limits
Understanding Torsional Constants
What is the Torsional Constant?
The torsional constant (K) is a geometric property that describes a cross-section's resistance to twisting. Unlike the polar moment of inertia, which only applies to circular sections, the torsional constant can be calculated for any cross-sectional shape.
Key Applications
- •Angle of twist calculations
- •Torsional stiffness analysis
- •Structural beam design
- •Aircraft wing analysis
Angle of Twist Formula
φ = TL / (KG)
- φ: Angle of twist (radians)
- T: Applied torque (N⋅m)
- L: Length of member (m)
- K: Torsional constant (m⁴)
- G: Shear modulus (Pa)
Important Assumptions
- Straight beam: Uniform cross-section throughout
- End loading: Torques applied only at ends
- Free warping: End sections can deform freely
- Elastic behavior: Stress below elastic limit
Torsional Constant Formulas
Circle
K = πr⁴/2
r = radius
Solid Rectangle
K = ab³/3 - 0.21b⁴ - 0.0175b⁸/a⁴
a ≥ b (larger dimension)
Solid Square
K = 9a⁴/64
a = side length
Solid Ellipse
K = πa³b³/(a²+b²)
a, b = semi-axes
Thin-Walled Ellipse
K = 2πa²b²t³/3(a²+b²)
t = wall thickness
Thin-Walled Rectangle
K = 2t³a²b²/(a+b)
t = wall thickness
Hollow Ellipse
K = K_outer - K_inner
Subtract inner from outer
I-Beam
K = 2K₁ + K₂ + 2αD⁴
Complex formula (see example)