U_max. = W/[pi]rt, T_max. = W/2[pi]rt.
The greatest value of U is in this case approximately twice its
average value, but it is possible that these results for the bending
of very thin tubes may be seriously at fault if the tube is not
plugged, and if the load is not applied in the manner contemplated in
the theory (cf. S 55). In such cases the extensions and contractions
of the longitudinal filaments may be practically confined to a small
part of the material near the ends of the tube, while the rest of the
tube is deformed without stretching.
51. The tangential tractions U, T on the cross-sections are necessarily
accompanied by tangential tractions on the longitudinal sections, and on
each such section the tangential traction is parallel to the central
line; on a vertical section z = const. its amount at any point is T, and
on a horizontal section y = const. its amount at any point is U.
The internal stress at any point is completely determined by the
components P, U, T, but these are not principal stresses (S 7). Clebsch
has given an elegant geometrical construction for determining the
principal stresses at any point when the values of P, U, T are known.
[Illustration: FIG. 14.]
From the point O (fig. 14) draw lines OP, OU, OT, to represent the
stresses P, U, T at O, on the cross-section through O, in magnitude,
direction and sense, and compound U and T into a resultant represented
by OE; the plane EOP is a principal plane of stress at O, and the
principal stress at right angles to this plane vanishes. Take M the
middle point of OP, and with centre M and radius ME describe a circle
cutting the line OP in A and B; then OA and OB represent the
magnitudes of the two remaining principal stresses. On AB describe a
rectangle ABDC so that DC passes through E; then OC is the direction
of the principal stress represented in magnitude by OA, and OD is the
direction of the principal stress represented in magnitude by OB.
[Illustration: FIG. 15.]
52. As regards the strain in the beam, the longitudinal and lateral
extensions and contractions depend on the bending moment in the same way
as in the simpler problem; but, the bending moment being variable, the
anticlastic curvature produced is also variable. In addition to these
extensions and contractions there are shearing strains corresponding to
the shearing stresses T, U. The shearing strain corresponding to T
consists of a relative sliding parallel to the central-line of different
longitudinal linear elements combined with a relative sliding in a
transverse horizontal direction of elements of different cross-sections;
the latter of these is concerned in the production of those
displacements by which the variable anticlastic curvature is brought
about; to see the effect of the former we may most suitably consider,
for the case of an elliptic cross-section, the distortion of the shape
of a rectangular portion of a plane of the material which in the natural
state was horizontal; all the boundaries of such a portion become
parabolas of small curvature, which is variable along the length of the
beam, and the particular effect under consideration is the change of the
transverse horizontal linear elements from straight lines such as HK to
parabolas such as H'K' (fig. 15); the lines HL and KM are parallel to
the central-line, and the figure is drawn for a plane above the neutral
plane. When the cross-section is not an ellipse the character of the
strain is the same, but the curves are only approximately parabolic.
The shearing strain corresponding to U is a distortion which has the
effect that the straight vertical filaments become curved lines which
cut the longitudinal filaments obliquely, and thus the cross-sections do
not remain plane, but become curved surfaces, and the tangent plane to
any one of these surfaces at the centroid cuts the central line
obliquely (fig. 16). The angle between these tangent planes and the
central-line is the same at all points of the line; and, if it is
denoted by 1/2[pi] + s0, the value of s0 is expressible as
shearing stress at centroid
---------------------------,
rigidity of material
and it thus depends on the shape of the cross-section; for the elliptic
section of S 50 its value is
4W 2a^2(1 + [sigma]) + b^2
------- -----------------------;
E[pi]ab 3a^2 + b^2
for a circle (with [sigma] = 1/4) this becomes 7W/2E[pi]a^2. The
vertical filament through the centroid of any cross-section becomes a
cubical parabola, as shown in fig. 16, and the contour lines of the
curved surface into which any cross-section is distorted are shown in
fig. 17 for a circular section.
[Illustration: FIG. 16.]