Torsional Rigidity
The angular twist theta of a shaft with given cross section is given by
| [画像: theta=(TL)/(KG) ] |
(1)
|
(Roark 1954, p. 174), where T is the twisting moment (commonly measured in units of inch-pounds-force), L is the length (inches), G is the modulus of rigidity (pounds-force per square inch), and K (sometimes also denoted C) is the torsional rigidity multiplier for a given geometric cross section (inches to the fourth power). Note that the quantity TL is sometimes denoted M_t (e.g., Timoshenko and Goodier 1951, p. 264).
Values of K are known exactly only for a small number of cross sections, and in closed form for even fewer. The following table lists approximate values for some common shapes (Timoshenko and Goodier 1951, pp. 258-280; Roark 1954, pp. 174-179).
Closed forms are known for the annulus
| K_(annulus)=1/2pi(a^4-b^4) |
(2)
|
(Roark 1954, p. 175), circle
| K_(circle)=1/2pia^4 |
(3)
|
(Roark 1954, p. 174), ellipse
(Timoshenko and Goodier 1951, p. 263-265; Roark 1954, p. 174), equilateral triangle
| K_(eq. tri.)=1/(80)sqrt(3)a^4 |
(5)
|
(Timoshenko and Goodier 1951, p. 265-267; Roark 1954, p. 175), and half-disk and slit full disk (i.e., circular sector from 0 to 2pi)
(E. Weisstein, Aug. 27, 2010; given approximately by Saint-Venant 1878ab; Timoshenko and Goodier 1951, p. 263-265; Roark 1954, p. 174).
Exact solutions expressed as sums (with no known closed form) are known for the rectangle and square
![画像:(a^3b)/3[1-(192)/(pi^5)a/bsum_(n=1)^(infty)1/((2n-1)^5)tanh((pi(2n-1)b)/(2a))]](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/TorsionalRigidity/Inline20.svg){kind=link}
![画像:(a^4)/3[1-(192)/(pi^5)sum_(n=1)^(infty)1/((2n-1)^5)tanh((pi(2n-1))/2)]](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/TorsionalRigidity/Inline23.svg){kind=link}
(Timoshenko and Goodier 1951, pp. 275-277), isosceles right triangle
![画像: K_(isos. rt. tri.)=a^4[1/(12)-(16)/(pi^5)sum_(n=1)^infty1/((2n-1)^5)coth((pi(2n-1))/2)]](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/TorsionalRigidity/NumberedEquation6.svg){kind=link}
(Galerkin 1919; correcting the typo 1/2 for 1/12), and circular sector
where
| f(r,psi)=-r^2[1-(cos(2psi))/(cosalpha)]+(16a^2alpha^2)/(pi^3)sum_(n=1,3,5,...)(-1)^((n+1)/2)(r/a)^(npi/alpha)(cos((npipsi)/alpha))/(n(n+(2alpha)/pi)(n-(2alpha)/pi)) |
(12)
|
(Saint-Venant 1878ab; Greenhill 1879; Dinnik, and Föppl and Föppl 1928; Timoshenko and Goodier 1951, pp. 278-280).
See also
Area Moment of Inertia, Radius of GyrationExplore with Wolfram|Alpha
References
Dinnik, A. Bull. Don. Polytech. Inst. Vovotcherkassk 1, 309.Föppl, A. and Föppl, L. Drang und Zwang. Munich, Germany: Oldenbourg, p. 96, 1928.Galerkin, B. G. "Torsion of a Triangular Prism." Izv. Ross. Akad. Nauk, Ser. 6 13, 111-118, 1919.Greenhill, A. G. "Fluid Motion in a Rotating Rectangle, Formed by Two Concentric Circular Arcs and Two Radii." Messenger Math. 9, 35-39, 1879.Roark, R. J. Formulas for Stress and Strain, 3rd ed. New York: McGraw-Hill, 1954.Saint-Venant, A.-J.-C. B. de. "Sur la torsion des prismes à base mixtiligne et sur une singularité que peuvent offrir certains emplois de la coordonnée logarithmique du système cylindrique isotherme de Lamé." C. R. Acad. Sci. Paris 87, 849-854, 1878a.Saint-Venant, A.-J.-C. B. de. "Exemples du calcul de la torsion de prismes à base mixtiligne." C. R. Acad. Sci. Paris 87, 893-898, 1878b.Sloane, N. J. A. Sequences A019669, A180309, A180310, A180311, A180314, and A180317 in "The On-Line Encyclopedia of Integer Sequences."Timoshenko, S. and Goodier, J. N. Theory of Elasticity, 2nd ed. New York: McGraw-Hill, 1951.Referenced on Wolfram|Alpha
Torsional RigidityCite this as:
Weisstein, Eric W. "Torsional Rigidity." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TorsionalRigidity.html