734
ELASTIC
section comes to its proper place on the curved central line ; (2) a rotation of the plane of each cross-section about the axis through its centroid at right angles to the plane of flexure, of such an amount as to place it at right angles to the curved central line; (3) a distortion of the shape of each cross-section in its own plane producing the anticlastic curvature. 2. That this theory requires modification when the load does not consist simply of terminal couples can be seen most easily by considering the problem of a beam loaded at one end with a weight W, and supported in a horizontal position at its other end. The forces that are exerted at any section y>, to balance the weight W, must reduce statically to a vertical force W and a couple, and these forces arise from the action of the part Ap on the part (see Fig. 2), i.e., from the stresses across the section at p. The stress that suffices in the simpler problem gives rise to no vertical force, and it is clear that in addition to longitudinal tensions and pressures there must be shearing stresses at the cross-sections. The determination of the character of these, and of the corresponding strains and displacements, was effected by Saint-Venant and Clebsch for a number of forms of sections by means of an analysis of the same kind as that employed in the solution of the torsion problem.
8. 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 is W(£ - x), and the longitudinal tension P at any point on the cross-section is -W(Z~x)yj, and this is related to the bending moment exactly as in the simpler problem. 4. 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 &(Fig. 3), the a axis being vertical in the unstrained state, and drawing the axis z at right angles to the plane of flexure, the vertical shearing stress U at any point (y, z) on any cross-section is 2W[(a2 - y2) {2a2(l + g) + b2} - zWjl - 2a)]. Tra3b(l + cr)(3a2 + 62) The resultant of these stresses is W, but the y amount at the centroid, which is the maximum amount, exceeds the average amount, Vf/irab, Fig. 3. in2 the ratio 2 {4a (l + <r) + 26 }/(3a2 + 62)(1 + <r). Taking cr = this ratio is -t for a circle, nearly f for a flat elliptic bar with the longest diameter vertical, nearly f for a flat elliptic bar with the longest diameter horizontal. In the same problem the horizontal shearing stress T at any point on any cross-section is of amount 4 Wyz {a2( 1 + a) + &V} ira3b(l + (r){3a2 + b2)
SYSTEMS The resultant of these stresses vanishes ; but, taking as before 0- = !, and putting for the three cases above a = b, a = lQb, b = 10a, the ratio of the maximum of this stress to the average vertical shearing stress has the values -§, nearly TV, 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 rv We find 4W P=- 7r(r 4 - TV1)' 0 U
= 2(l + ,MV-V)[(ii+2,){^+^-^
T= - U +
W
+
^+<3+2,,)(V+?),}r' ’ 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(Z - x)yl'irr3t Umal. = W/7rrq Tmax. = W/2rrrh 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. § 9). In such cases the extensions and contractions of the longitudinal fibres 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. 5. The shearing stresses U, T on the cross-sections are necessarily accompanied by shearing stresses on the longitudinal sections, and on each such section the shearing stress 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. Clebsch has given an elegant geometrical construction for determining the principal stresses at any point when the values of P, U, T are known. From the point 0 (Fig. 4) 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 U plane EOP is a principal plane of stress at 0, 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. 6. 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 isT concerned in the production of those displacements by w hich 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 crosssection the distortion of the shape of a rectangular portion
