T = − W{ 1 + 2σ + (3 + 2σ) r0² r1²} yz;
(1 + σ) π (r04 − r14)(y² + z²)²

and for a tube of radius r and small thickness t the value of P and the maximum values of U and T reduce approximately to

P = − W (l − x)y / πr³t

Umax. = W / πrt,   Tmax. = W / 2πrt.

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. § 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 (§ 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.

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.

Fig. 15.