L[root](W/EI) = 4.6, [alpha] = 130 deg.
approximately. It is noteworthy that whenever the length and force
admit of a sinuous form, such as [alpha] or b, with more than two
inflections, there is also possible a crossed form, like e, with two
inflections only; the latter form is stable and the former unstable.
61. The particular case of the above for which [alpha] is very small is
a curve of sines of small amplitude, and the result in this case has
been applied to the problem of the buckling of struts under thrust. When
the strut, of length L', is maintained upright at its lower end, and
loaded at its upper end, it is simply contracted, unless L'^2W >
1/4[pi]^2EI, 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 1/4[pi]^2EI, 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. S
56).
[Illustration: FIG. 25.]
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 > [pi]^2EI; 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 >
4[pi]^2EI, the central-line forming a complete bay and two half bays
(fig. 25, c).
[Illustration: FIG. 26.]
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 SS 48-54, and supposing
that one principal axis of a cross-section at its centroid makes an
angle [theta] 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).
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 (S 42). The twist [tau] 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 [tau]v. 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.
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 1/4E[pi]c^4/(1 + [sigma]); and the
flexural rigidity, which is the same for all planes through the
central-line, is 1/4E[pi]c^4; 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 [tau], and then the
torsional couple is C[tau], and the flexural couple in the osculating
plane is A/[rho], where [rho] 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 [alpha] be the angle and r the radius of the helix, so that [rho]
is r sec^2[alpha]; 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
sin [alpha] cos^3 [alpha] cos [alpha]
R = A ------------------------- - C[tau] ------------,
r^2 r
cos^3 [alpha]
K = A ------------- + C[tau] sin [alpha].
r
The thrust across any section is R sin [alpha] parallel to the tangent
to the helix, and the shearing stress-resultant is R cos [alpha] at
right angles to the osculating plane.
[Illustration: FIG. 27.]
When the twist is such that, if the rod were simply unbent, it would
also be untwisted, [tau] is (sin [alpha] cos [alpha])/r, and then,
restoring the values of A and C, we have
E[pi]c^4 [sigma]
R = -------- ------------ sin [alpha] cos^2 [alpha],
4r^2 1 + [sigma]
E[pi]c^4 1 + [sigma] cos^2 [alpha]
K = -------- ------------------------- cos [alpha].
4r 1 + [sigma]
65. The theory of spiral springs affords an application of these
results. The stress-couples called into play when a naturally helical
spring ([alpha], r) is held in the form of a helix ([alpha]', r'), are
equal to the differences between those called into play when a
straight rod of the same material and section is held in the first
form, and those called into play when it is held in the second form.