but the direction of the traction at a point of the cross-section
need not in general be vertical. The existence of tangential
traction on the cross-sections implies the existence of equal
tangential traction, directed parallel to the central-line, on
some planes or other which are parallel to this line, the two sets
of tractions forming a shearing stress. We conclude that such
shearing stress is a necessary constituent of the stress-system
in the beam bent by terminal transverse load. We can develop
a theory of this stress-system from the assumptions (i.) that the
tension at any point of the cross-section is related to the bending
moment at the section by the same law as in the case of uniform
bending by terminal couples; (ii.) that, in addition to this
tension, there is at any point shearing stress, involving tangential
tractions acting in appropriate directions upon the elements
of the cross-sections. When these assumptions are made it
appears that there is one and only one distribution of shearing
stress by which the conditions of the problem can be satisfied.
The determination of the amount and direction of this shearing
stress, and of the corresponding strains and displacements, was
effected by Saint-Venant and R. F. A. Clebsch for a number of
forms of section by means of an analysis of the same kind as that
employed in the solution of the torsion problem.
 |
| Fig. 13. |
49. Let l be the length of the beam, x the distance of the section p from the fixed end A, y the distance of any point below the horizontal plane through the centroid of the section at A, then the bending moment at p is W (l − x), and the longitudinal tension P or Xx at any point on the cross-section is −W (l − x)y/I, and this is related to the bending moment exactly as in the simpler problem.
50. The expressions for the shearing stresses depend on the shape of the cross-section. Taking the beam to be of isotropic material and the cross-section to be an ellipse of semiaxes a and b (fig. 13), the a axis being vertical in the unstrained state, and drawing the axis z at right angles to the plane of flexure, we find that the vertical shearing stress U or Xy at any point (y, z) on any cross-section is
| 2W [(a2 − y2) {2a2 (1 + σ) + b2} − z2a2 (1 − 2σ)] | . |
| πa3b (1 + σ) (3a2 + b2) |
The resultant of these stresses is W, but the amount at the centroid, which is the maximum amount, exceeds the average amount, W/πab, in the ratio
If σ = 14, this ratio is 75 for a circle, nearly 43 for a flat elliptic bar with the longest diameter vertical, nearly 85 for a flat elliptic bar with the longest diameter horizontal.
In the same problem the horizontal shearing stress T or Zx at any point on any cross-section is of amount
| − | 4Wyz {a2 (1 + σ) + b2σ} | . |
| πa3b (1 + σ) (3a2 + b2) |
The resultant of these stresses vanishes; but, taking as before σ = 14, and putting for the three cases above a = b, a = 10b, b = 10a, we find that the ratio of the maximum of this stress to the average vertical shearing stress has the values 35, nearly 115, and nearly 4. Thus the stress T is of considerable importance when the beam is a plank.
As another example we may consider a circular tube of external radius r0 and internal radius r1. Writing P, U, T for Xx, Xy, Zx, we find
| P = − | 4W | (l − x)y, |
| π (r04 − r14) |
| U = | W | [ (3 + 2σ) { r02 + r12 − y2 − | r02 r12 | (y2 − z2) } − (1 − 2σ) z2 ] |
| 2(1 + σ) π (r04 − r14) | (y2 + z2)2 |
| 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
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. |
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 12π + s0, the value of s0 is expressible as
| shearing stress at centroid | , |
| rigidity of material |
