maintained upright at its lower end, and loaded at its upper end, it is simply contracted, unless L′2W > 14π2EI, for the lower end corresponds to a point at which the tangent is vertical on an elastica for which the line of inflections is also vertical, and thus the length must be half of one bay (fig. 25, a). For greater lengths or loads the strut tends to bend or buckle under the load. For a very slight excess of L′2W above 14π2EI, the theory on which the above discussion is founded, is not quite adequate, as it assumes the central-line of the strut to be free from extension or contraction, and it is probable that bending without extension does not take place when the length or the force exceeds the critical value but slightly. It should be noted also that the formula has no application to short struts, as the theory from which it is derived is founded on the assumption that the length is great compared with the diameter (cf. § 56).
The condition of buckling, corresponding to the above, for a long strut, of length L′, when both ends are free to turn is L′2W > π2EI; for the central-line forms a complete bay (fig. 25, b); if both ends are maintained in the same vertical line, the condition is L′2W > 4π2EI, the central-line forming a complete bay and two half bays (fig. 25, c).
62. In our consideration of flexure it has so far been supposed that the bending takes place in a principal plane. We may remove this restriction by resolving the forces that tend to produce bending into systems of forces acting in the two principal planes. To each plane there corresponds a particular flexural rigidity, and the systems of forces in the two planes give rise to independent systems of stress, strain and displacement, which must be superposed in order to obtain the actual state. Applying this process to the problem of §§ 48-54, and supposing that one principal axis of a cross-section at its centroid makes an angle θ with the vertical, then for any shape of section the neutral surface or locus of unextended fibres cuts the section in a line DD′, which is conjugate to the vertical diameter CP with respect to any ellipse of inertia of the section. The central-line is bent into a plane curve which is not in a vertical plane, but is in a plane through the line CY which is perpendicular to DD′ (fig. 26).
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| Fig. 26. |
63. Bending and Twisting of Thin Rods.—When a very thin rod or wire is bent and twisted by applied forces, the forces on any part of it limited by a normal section are balanced by the tractions across the section, and these tractions are statically equivalent to certain forces and couples at the centroid of the section; we shall call them the stress-resultants and the stress-couples. The stress-couples consist of two flexural couples in the two principal planes, and the torsional couple about the tangent to the central-line. The torsional couple is the product of the torsional rigidity and the twist produced; the torsional rigidity is exactly the same as for a straight rod of the same material and section twisted without bending, as in Saint-Venant’s torsion problem (§ 42). The twist τ is connected with the deformation of the wire in this way: if we suppose a very small ring which fits the cross-section of the wire to be provided with a pointer in the direction of one principal axis of the section at its centroid, and to move along the wire with velocity v, the pointer will rotate about the central-line with angular velocity τv. The amount of the flexural couple for either principal plane at any section is the product of the flexural rigidity for that plane, and the resolved part in that plane of the curvature of the central line at the centroid of the section; the resolved part of the curvature along the normal to any plane is obtained by treating the curvature as a vector directed along the normal to the osculating plane and projecting this vector. The flexural couples reduce to a single couple in the osculating plane proportional to the curvature when the two flexural rigidities are equal, and in this case only.
The stress-resultants across any section are tangential forces in the two principal planes, and a tension or thrust along the central-line; when the stress-couples and the applied forces are known these stress-resultants are determinate. The existence, in particular, of the resultant tension or thrust parallel to the central-line does not imply sensible extension or contraction of the central filament, and the tension per unit area of the cross-section to which it would be equivalent is small compared with the tensions and pressures in longitudinal filaments not passing through the centroid of the section; the moments of the latter tensions and pressures constitute the flexural couples.
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| Fig. 27. |
64. We consider, in particular, the case of a naturally straight spring or rod of circular section, radius c, and of homogeneous isotropic material. The torsional rigidity is 14Eπc4 / (1 + σ); and the flexural rigidity, which is the same for all planes through the central-line, is 14Eπc4; we shall denote these by C and A respectively. The rod may be held bent by suitable forces into a curve of double curvature with an amount of twist τ, and then the torsional couple is Cτ, and the flexural couple in the osculating plane is A/ρ, where ρ is the radius of circular curvature. Among the curves in which the rod can be held by forces and couples applied at its ends only, one is a circular helix; and then the applied forces and couples are equivalent to a wrench about the axis of the helix.
Let α be the angle and r the radius of the helix, so that ρ is r sec2α; and let R and K be the force and couple of the wrench (fig. 27).
Then the couple formed by R and an equal and opposite force at any section and the couple K are equivalent to the torsional and flexural couples at the section, and this gives the equations for R and K
| R = A | sin α cos3 α | − | cos α | , |
| r2 | r |
| K = A | cos3 α | + Cτ sin α. |
| r |
The thrust across any section is R sin α parallel to the tangent to the helix, and the shearing stress-resultant is R cos α at right angles to the osculating plane.
When the twist is such that, if the rod were simply unbent, it
