The diameter of the distorted cross-sections does not vary with z, hence the radial component of the displacement is a function of the radius only, u = u(r). Assuming u
rr r xa
Fig. 1.50 Torsion of a rod by opposite twisting moments applied at its ends that in the torsion the element volume does not change even for strain terms of quadratic form such as T2a2, from (1.122c) £rr + £tt + £zz = 0. From (1.121b), this entails dl + u 1 t2„2 dr r
We obtain u = 1 T2a2(r + c1/r). Setting the constant c1 = 0 for a plain rod, u = 1 T2a2r, (1.130b)
which shows that the radius of the cross-section is increased.
Any point located in a cross-section rotates by the same angle 9 = Ta; therefore a straight radial line of the cross-section is assumed to remain straight. At the distance z from the origin, the length of the curvilinear displacement is v = Taz, and for any point of the rod this displacement is represented by v = T rz. (1.131)
The shear strain etz purely applies along circles centered on the z-axis which, for any point at distance r = a, leads to etz = Ta/2. Finally the strain and displacement components are
The strain-stress relations (1.111a) with the subscripts x, y, z substituted by r, t, z provide the determination of the stress components at a current point r = a. Using only the shear modulus instead of the Young modulus and Poisson's ratio [cf. Eq. (1.117)], we obtain after simplification azz = — G T2a2, = G Ta. (1.133)
One may notice that the stress component ozz varies with the square power of the torsion angle. This entails that the effect of this stress with respect to the shearing stress component otz becomes important when the T-value is large.
Let now a be the external radius of the rod. If the central length of the rod were unchanged, the axial stress would be null at the center of the cross-section. But from (1.128) the shortening w = —T2a2z/2 of the rod axis is due to a compression of the central zone. Hence the axial stress distribution or stress function oz(r) must vary as r2/a2 over the cross-section and must have a finite value on the axis. An appropriate representation is of the form az(r) = azz (Cx — r2/a2), (1.134)
where C1 is a constant. Since no axial force is applied at the rod ends, the resultant of the stress elements taken over the cross-section must be zero in this direction,
which gives C = 1/2. The stress distribution is
and takes a null value at the distance r/a = 1/from the axis. The axial stresses at the center of the cross-section is a compression and is opposite at the edge, oz(0) = -Oz(a) = -Gr2a2/2. (1.136b)
Since only the moments are applied to the ends of the rod, we assume that the rod is long enough so the stress distribution o(r) arises in the main part of the rod and locally vanishes at the ends.
• Torsion with constant length: From the above result, we conclude that the diameter and length of the rod may be made unchanged for a large torsion angle if, in addition to the opposite twisting moments Mt, opposite external forces Fz are applied to the ends (cf. Fig. 1.50).
The intensity of the stretching force Fz to apply towards the positive z-end is derived from oz(0) in (1.136b), fa
therefore the rod is elongated and recovers its initial length.
1.13.6 Love-Kirchhoff Hypotheses and Thin Plate Theory
The bending of plates is a complex three-dimensional problem which can only be accurately solved from finite element analysis. For the basic and classical case where the thickness t of the plate is small compared to the typical overall dimension I of its two-dimension surface area, the so-called thin plate theory allows reducing the problem to a two-dimension one by introducing some conditions. These conditions are usually referred to as Love-Kirchhoff hypotheses.
Assuming a plane plate and setting an x, y plane that lies at its middle plane and z-axis normal to this plane, Love-Kirchhoff hypotheses are as follows.
1. For points originally at the mid-plane, the components of the displacement vector reduce to u |z=0 = v |z=0 = 0, w z=0 = w(x,y). (1.138a)
2. Any point located on normal lines to the mid-plane remains on normal lines to the deformed mid-surface (known as Euler-Bernoulli hypothesis). No stress exists along these normal lines,
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