SOME SIMPLE STRESS-DISTRIBUTIONS Turning from the methods to the results of mathematical enquiry, we remark that some of its simplest solu tions are those which have proved most useful to the practical engineer. Thus the "tie," or member subjected to tensile load, is one of the commonest of structural units : theory states that the stress in a straight tie of uniform cross-section will be a simple tensile stress of uniform intensity, provided that the line of the resultant pull passes through the centroid, or centre of area, of each cross-section. At the ends this simple stress-distribution will be modified according to the manner in which the load is applied; but theory shows that the modification is unimportant, except in the immediate neighbourhood of the ends. Thus, when we subject a straight rod of material to a tensile test, it does not matter how the load is applied, provided that the line of the pull is accurately central, and that the resulting strain is measured over a limited part of the length, remote from the loaded ends.
Another structural unit of very common occur rence is the beam or girder, a member whose function is to resist bending. B. de St. Venant (1856) first put the theory of bending on a satisfactory basis (J. de Math. [Liouville], Ser. 2, vol.
I). He showed that a state of stress can be maintained in a straight beam of uniform section, such that there is at every point a purely longitudinal stress (either tensile or compressive). The intensity of this stress depends upon the position of the point : in any cross-section of the beam (fig. 8) there will be a line NN such that at points on this line the stress is zero ; at any other point P, the intensity of stress will be proportional to y, the dis tance of P from NN. NN will always pass through the centroid
of the cross-section : if this has a plane of symmetry (QQ in the figure) which is also the plane of bending, then NN will be per pendicular to that plane.
Let R be the radius of the (very large) circle into which the beam is bent ; let I be the "geometrical moment of inertia" of the cross-section about the line NN; and let E be Young's modulus for the material. Then the longitudinal stress p at P (acting at right angles to the cross-section) will be given by It should be emphasized that these formulae apply only to shafts of circular cross-section.
Principle of Superposition. Bending Combined with A consequence follows from the fact that the elastic equations are linear in form. If we can calculate the stresses, strains or displacements which result when any two systems of load act separately on a given body, then we know that these two systems, acting simultaneously, will produce stresses, strains or where a is the radius of the shaft (assumed solid).
Stresses Due to Internal Pressure, in Thin Circular Tubes or Spherical Shells.-34. When a long circular tube of uniform thickness is subjected to uniform internal pressure, considerations of symmetry show that the stresses exerted across any plane which contains the axis will be purely normal ; moreover, if the thickness is small, this normal stress must be practically uniform. Let t be the thickness of the tube, r the inner radius, P the internal pressure, and p the stress; then, considering any length 1 of the tube, we see that a total force due to pressure, of amount This result has been utilized in the experimental determination of k (see § 48).