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Torsional Rigidity


The angular twist theta of a shaft with given cross section is given by

 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).

cross sectionK/a^4 approxOEIS
circle1.570796...A019669
equilateral triangle0.021650...A180317
half-disk0.297556...A180310
isosceles right triangle0.026089...A180314
quarter-disk0.0825...
sliced disk0.878055...A180311
square0.140577...A180309

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

 K_(ellipse)=(pia^3b^3)/(a^2+b^2)
(4)

(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)

K_(half-disk)=a^4(pi/2-4/pi)
(6)
K_(full-disk)=a^4(pi-(64)/(9pi))
(7)

(E. Weisstein, Aug. 27, 2010; given approximately by Saint-Venant 1878; 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

K_(rectangle)=(a^3b)/3[1-(192)/(pi^5)a/bsum_(n=1)^(infty)1/((2n-1)^5)tanh((pi(2n-1)b)/(2a))]
(8)
K_(square)=(a^4)/3[1-(192)/(pi^5)sum_(n=1)^(infty)1/((2n-1)^5)tanh((pi(2n-1))/2)]
(9)

(Timoshenko and Goodier 1951, pp. 275-277), isosceles right triangle

 K_(isos. rt. tri.)=a^4[1/(12)-(16)/(pi^5)sum_(n=1)^infty1/((2n-1)^5)coth((pi(2n-1))/2)]
(10)

(Galerkin 1919; correcting the typo 1/2 for 1/12), and circular sector

 K_(circ. sector)=int_0^aint_(-alpha/2)^(alpha/2)f(r,psi)dpsidr
(11)

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 1878; Greenhill 1879; Dinnik, and Föppl and Föppl 1928; Timoshenko and Goodier 1951, pp. 278-280).


See also

Area Moment of Inertia, Radius of Gyration

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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. Akad. Nauk, SSSR pp. 111-118, 1919.Greenhill, A. G. Messenger Math. 9, 35, 1879.Roark, R. J. Formulas for Stress and Strain, 3rd ed. New York: McGraw-Hill, 1954.Saint-Venant. Compt. Red. 87, 849 and 903, 1878.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.

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Torsional Rigidity

Cite this as:

Weisstein, Eric W. "Torsional Rigidity." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TorsionalRigidity.html

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