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Beyond Circles: What is the mystery of torsion constants in different shapes?
The torsion constant or torsion coefficient is the geometric property of the cross-section of a bar material.It involves the relationship between the two, the torsional angle of the bar material, which is very important in homogeneous linear elastic bar material.Tortment constant, together with material properties and length, describes the torsional stiffness of a bar material, and its international unit is m4.
History
As early as 1820, French engineer A. Duleau concluded through analytical derivation that the torsion constant of the beam is equal to the second moment Jzz normal to the cross section.This theorem is based on the assumption that the plane section before the twist remains flat after the twist, and the diameter line will not change.However, this assumption is only true in beams with circular cross-sections and does not apply to any other shapes where warping occurs.For non-circular cross-sections, there is no exact analytical equation to calculate the torsion constant, but approximate solutions to many shapes have been found.Non-circular cross-sections are always accompanied by warping and deformation, and numerical methods are required to perform accurate torsion constant calculations.If warping of the end section is restricted by, for example, a rigid end block, the torsional stiffness of the non-circular cross-section beam can be significantly increased.
Formation of torsion constant
For beams with uniform length cross-sections, the torsion angle (indicated in radians) can be expressed by the following relationship:
θ = T * L / (G * J)
Where, T represents the applied torque, L is the length of the beam, G is the stiffness modulus (shear modulus) of the material, and J is the torsion constant.In reverse, we can define two quantities, namely torsional stiffness GJ and torsional stiffness GJ/L.
Example
These shapes are special cases when we consider bar materials with specific uniform cross-sectional shapes.
Circular
For circular cross-sections, Jzz = (π * r^4) / 2
This formula shows that when the radius is r, it is equivalent to the exact representation of the second moment Jzz.
Oval
For elliptical cross-sections, J ≈ (π * a^3 * b^3) / (a^2 + b^2)
Here a is the large radius and b is the small radius.
Square
For square cross-section, J ≈ 2.25 * a^4
Here a is half the length of the side.
Rectangle
For rectangular cross-sections, J ≈ β * a * b^3, where β is determined according to a specific table.
Here a is the long side and b is the short side, which helps to understand the effects of different proportions.
Thin-walled open round tube
The torsion constant of such cross-sections is J = (1/3) * U * t^3, where U is the length of the median boundary and t is the wall thickness.
Round thin-walled open tube
At this time J = (2/3) * π * r * t^3, where t is the wall thickness and r is the average radius.
In summary, although in the case of circles and other simple geometric shapes, we can use precise formulas to calculate the torsion constant, the required methods become increasingly cumbersome as the complexity of the shape increases.Does this mean that the future of engineering design needs to consider more complex geometric models for optimal results?
