Abstract
In this chapter we shall consider the Saint Venant theory of torsion for uniform prismatic elastic rods loaded by twisting couples at the ends of the rod (see Fig. 5.1). You will recall that for the special case of a circular cross section, it is assumed in strength of materials (and shown valid in the theory of elasticity) that cross sections of the rod merely rotate as rigid surfaces under the action of the twisting couples.
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Notes
- 1.
Recall we have done this for the case of plane stress in Sec. 3.4.
- 2.
We can now see from the following equation that specifying the torques M at the ends of the shaft is equivalent to specifying the rate of twist α. That is, M and α play equivalent roles in the mathematical aspects of the problem.
- 3.
See Timoshenko and Goodier: “Theory of Elasticity,” McGraw-Hill Book Co., Chap. 11.
- 4.
- 5.
Recall from Problem 1.8 that:
$${\tau _{{\rm{oct}}}}^2 = \frac{1}{9}[2{({I_\tau })^2} + 6{(I{I_\tau })^2}$$From this formulation we see that:
$${\tau _{{\rm{oct}}}} = \frac{1}{3}\sqrt {{{({\tau _{xx}} - {\tau _{yy}})}^2} + {{({\tau _{xx}} - {\tau _{zz}})}^2} + {{({\tau _{yy}} - {\tau _{zz}})}^2} + 6\left( {{\tau _{xy}}^2 + {\tau _{xz}}^2 + {\tau _{yz}}^2} \right)}$$Similarly ε oct is given as:
$${\varepsilon _{{\rm{oct}}}} = \frac{1}{3}\sqrt {{{({\varepsilon _{xx}} - {\varepsilon _{yy}})}^2} + {{({\varepsilon _{xx}} - {\varepsilon _{zz}})}^2} + {{({\varepsilon _{yy}} - {\varepsilon _{zz}})}^2} + \frac{3}{2}\left( {{\gamma _{xy}}^2 + {\gamma _{yz}}^2 + {\gamma _{xz}}^2} \right)}$$ - 6.
For a more detailed explanation of this step as well as the problem of inelastic torsion under discussion see Smith and Sidebottom: “Inelastic Behavior of Load Carrying Members,” John Wiley and Sons, Chaps. 2–8.
- 7.
A procedure for accomplishing this effectively is developed in Smith and Sidebottom: “Inelastic Behavior of Load Carrying Members,” John Wiley and Sons, Chap. 1.
- 8.
See Mikhlin: “Variational Methods in Mathematical Physics,” The Macmillan Co., 1964.
- 9.
See Sokolnikoff: “Mathematical Theory of Elasticity,” McGraw-Hill Book Co., 1956, pp. 427, 428.
- 10.
That is, if we can approximate any harmonic function arbitrarily closely at all but isolated points by the sum
$$\sum\limits_{i = 1}^n {{a_i}{v_i}}$$by properly choosing the a’s and making n large enough.
- 11.
Remember D = M/α. It is clear that M and α must have the same sense and so M/α must be, positive.
- 12.
Such a set of polynomials can be shown to be complete.
- 13.
Note that by this process the problem has been altered such as to require the solution of a Galerkin integral (see Sec. 3.15). We shall make use of this transition in the Kantorovich method at later times in the text.
- 14.
L. V. Kantorovich and V. I. Krylov: “Approximate Methods of Higher Analysis,” Interscience, N.Y., 1964.
- 15.
Arnold Kerr, “An Extension of the Kantorovich Method,” Quarterly of Applied Mathematics, July, 1968.
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Dym, C.L., Shames, I.H. (2013). Torsion. In: Solid Mechanics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6034-3_5
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