torsion(redirected from Torsion (mathematics))
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torsion,stress on a body when external forces tend to twist it about an axis. See strength of materialsstrength of materials,
measurement in engineering of the capacity of metal, wood, concrete, and other materials to withstand stress and strain. Stress is the internal force exerted by one part of an elastic body upon the adjoining part, and strain is the deformation or change in
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(in strength of materials), a type of deformation characterized by rotation of the cross sections of a rod or shaft relative to one another under the influence of moments (force couples) acting in them. Upon torsion, the cross sections of round rods or shafts remain plane; upon torsion of a prismatic bar, warping of the cross section takes place (it does not remain plane). If the warping is different in various sections, then normal stresses also arise in the cross sections of the rod, in addition to tangential stresses. In this case the torsion is called constrained. In free torsion, in which the warping in all sections is identical, only tangential stresses arise in the cross section.
In practice, the most common type of torsion is torsion of a round, straight rod (Figure 1). As a result of the torsion moment Mt tangential stresses τ arise in the cross sections of the rod, and the rod segments (separated by distance l) are rotated relative to each other by an angle of twist ϕ. The angle of twist per unit length of the rod is called the relative angle of twist Θ. In free torsion, the relative angle of twist and the maximum tangential stresses τmax in the elastic phase are found from the formulas
θ = Mt/GIt τmax = Mt/Wt
where G is the elastic shear modulus and It and Wt are the arbitrary moment of inertia and the sectional modulus of torsion. In round sections, It is a polar moment of inertia Ip = πr4/2, and Wt is a polar moment of resistance Wp = πr3/2. For rectangular sections with a greater side h and lesser side b, It = αhb3 and Wt = βhb2, where the coefficients α and β are determined according to the ratio h/b from a table. For narrow sections (h/b ≧ 10), it is possible to take α = β ≈ 1/3.
For torsion of a round shaft in the elastic phase, the tangential stresses are distributed linearly throughout the cross section (Figure 2,a) and are determined by the formula τ = (Mt/ Iρ)ρ, where ρ is the distance from the axis of the shaft to the point of the section in question. In the elastic-plastic phase the torsional tangential stresses corresponding to the yield limit τy propagate from the surface toward the axis of the shaft (Figure 2,b). In the limiting state, the plastic zone extends to the axis of the shaft (Figure 2,c), and the limiting tortion moment for a round section is
Mlim = 2/3 (τyπr3)
The concept of torsion has also been extended to the action of internal tangential stresses arising during deformation of plates and casings.
REFERENCESBeliaev, N. M. Soprotivlenie materialov, 12th ed. Moscow, 1959.
Kurs soprotivleniia materialov, 5th ed., part 1. Moscow, 1961.
Novozhilov, V. V. Teoriia uprugosti. Leningrad, 1958.
L. V. KASAB’IAN
(mathematics), the measure of the deviation of a space curve from its osculating plane. The torsion of a curve at the point M is determined in the following manner. Let N be a variable point on a curve sufficiently close to M and let β be the acute angle between the osculating planes at M and N. The angle β is considered to be positive if as N approaches M a variable osculating plane appears to an observer looking from M to N to rotate in the counterclockwise direction; in the opposite case the angle β is considered to be negative. The limit of the ratio β/ΔS—where ΔS is the arc length MN —as N approaches M is called the torsion < of the curve at the point M: