Solutions to Particular Three-Dimensional Boundary Value Problems of Elastostatics
In this chapter the boundary value problems related to torsion of a prismatic bar bounded by a cylindrical lateral surface and by a pair of planes normal to the lateral surface, are discussed. It is assumed that a resultant torsion moment is applied at on
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Solutions to Particular Three-Dimensional Boundary Value Problems of Elastostatics
In this chapter the boundary value problems related to torsion of a prismatic bar bounded by a cylindrical lateral surface and by a pair of planes normal to the lateral surface, are discussed. It is assumed that a resultant torsion moment is applied at one of the bases while the other is subject to a warping and the lateral surface is stress free. In each of the problems an approximate three-dimensional formulation is reduced to a two-dimensional one for Laplace’s or Poisson’s equation on the cross section of the bar.
8.1 Torsion of Circular Bars We consider a circular prismatic bar of length l and radius a referred to the Cartesian coordinates (x1 , x2 , x3 ) in such a way that x3 coincides with the axis of the bar, the bar is fixed at x3 = 0 in the (x1 , x2 ) plane, while at x3 = l a torsion moment M3 is applied. This moment causes the bar to be twisted, and the generators of the circular cylinder deform into helical curves. An elastic state s = [u, E, S] in the bar is approximated by s = [ u, E, S], where u 2 = α x1 x3 , u3 = 0 u 1 = −α x2 x3 ,
(8.1)
and α is the angle of twist per unit length along the x3 axis
and
11 = E 22 = E 33 = E 12 = 0 E 31 = − 1 α x2 23 = 1 α x1 , E E 2 2
(8.2)
S11 = S22 = S33 = S12 = 0 S23 = μα x1 , S31 = −μα x2
(8.3)
M. Reza Eslami et al., Theory of Elasticity and Thermal Stresses, Solid Mechanics and Its Applications 197, DOI: 10.1007/978-94-007-6356-2_8, © Springer Science+Business Media Dordrecht 2013
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8 Solutions to Particular Three-Dimensional Boundary Value
The torsion moment M3 is M3 = (x1 (x12 + x22 )d x 1 d x 2 ≡ μ α J S23 − x2 S31 )d x 1 d x 2 = μα A
(8.4)
A
where A is the area of the cross section : A = {(x1 , x2 ) : x12 + x22 ≤ a}, and J is the polar moment of inertia of the cross section about its center. The product μJ is called the torsional rigidity of the bar. Also, since n α = xα /a, n 3 = 0 on ∂A (α = 1, 2)
(8.5)
therefore, because of Eq. (8.3) Si j n j = 0 on ∂A (i, j = 1, 2, 3)
(8.6)
that is, the elastic state s = [ u, E, S] satisfies the homogeneous boundary conditions: (i) u = 0 on x3 = 0, and (ii) Sn = 0 on the lateral surface ∂A × [0, l]. A shear stress boundary condition on the plane x3 = l is replaced by application of the resultant moment M3 on this plane. Torsion of Noncircular Prismatic Bars A noncircular prismatic bar of length l is fixed at x3 = 0 in the sense that the displacement components in the (x1 , x2 ) plane vanish while the axial displacement is subject to a warping, and the other end x3 = l is twisted by a moment M3 ; the lateral surface of the bar is stress free and no body forces are present. Therefore, an elastic state s = [u, E, S] in the bar is approximated by s = [ u, E, S] in which u 2 = α x1 x3 , u 3 = α ψ(x1 , x2 ) u 1 = −α x2 x3 ,
(8.7)
where ψ = ψ(x1 , x2 ) is called a warping function. The strain-displacement relations, the equilibrium equations with zero b
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