The torsion constant or torsion coefficient is a geometrical property of a bar's cross-section. It is involved in the relationship between angle of twist and applied torque along the axis of the bar, for a homogeneous linear elastic bar. The torsion constant, together with material properties and length, describes a bar's torsional stiffness. The SI unit for torsion constant is m4.
In 1820, the French engineer A. Duleau derived analytically that the torsion constant of a beam is identical to the second moment of area normal to the section Jzz, which has an exact analytic equation, by assuming that a plane section before twisting remains planar after twisting, and a diameter remains a straight line.Unfortunately, that assumption is correct only in beams with circular cross-sections, and is incorrect for any other shape where warping takes place.[1]
For non-circular cross-sections, there are no exact analytical equations for finding the torsion constant. However, approximate solutions have been found for many shapes.Non-circular cross-sections always have warping deformations that require numerical methods to allow for the exact calculation of the torsion constant.[2]
The torsional stiffness of beams with non-circular cross sections is significantly increased if the warping of the end sections is restrained by, for example, stiff end blocks.[3]
For a beam of uniform cross-section along its length, the angle of twist (in radians)
\theta
\theta=
| TL | |
| GJ |
T is the applied torque
L is the beam length
G is the modulus of rigidity (shear modulus) of the material
J is the torsional constant
Inverting the previous relation, we can define two quantities; the torsional rigidity,
GJ=
| TL | |
| \theta |
And the torsional stiffness,
| GJ | |
| L |
=
| T | |
| \theta |
Bars with given uniform cross-sectional shapes are special cases.
Jzz=Jxx+Jyy=
| \pir4 | |
| 4 |
+
| \pir4 | |
| 4 |
=
| \pir4 | |
| 2 |
r is the radiusThis is identical to the second moment of area Jzz and is exact.
alternatively write:
J=
| \piD4 | |
| 32 |
D is the Diameter
J ≈
| \pia3b3 | |
| a2+b2 |
a is the major radius
b is the minor radius
J ≈ 2.25a4
a is half the side length.
J ≈ \betaab3
a is the length of the long side
b is the length of the short side
\beta
| a/b | \beta | |
|---|---|---|
| 1.0 | 0.141 | |
| 1.5 | 0.196 | |
| 2.0 | 0.229 | |
| 2.5 | 0.249 | |
| 3.0 | 0.263 | |
| 4.0 | 0.281 | |
| 5.0 | 0.291 | |
| 6.0 | 0.299 | |
| 10.0 | 0.312 | |
infty | 0.333 |
Alternatively the following equation can be used with an error of not greater than 4%:
J ≈
| ab3 | |
| 16 |
\left(
| 16 | |
| 3 |
-{3.36}
| b | |
| a |
\left(1-
| b4 | |
| 12a4 |
\right)\right)
a is the length of the long side
b is the length of the short side
J=
| 1 | |
| 3 |
Ut3
t is the wall thickness
U is the length of the median boundary (perimeter of median cross section)
This is a tube with a slit cut longitudinally through its wall. Using the formula above:
U=2\pir
J=
| 2 | |
| 3 |
\pirt3
t is the wall thickness
r is the mean radius