On the Theory of Flexure.
By William H. Burr, Rensselaer Polytechnic Institute, Troy, N. Y.
It is not intended in this discussion to give the exact theory of flexure for all materials and shapes of pieces subjected to bending, nor indeed for any one kind of material. The present state of knowledge regarding the internal molecular action developed in any piece of elastic material by the action of external forces, is not such as to enable one to treat any problem of this kind with mathematical rigor if the piece be of finite dimensions. The illustrious Lamé, however, has remarked that the exact solutions of all problems in natural science are usually obtained by successive approximations, and such has certainly been the case in regard to the theory of flexure.
If the following investigation shall be found to constitute even a short step in the direction of the correct theory, the object of the writer will have been accomplished.
*An explanation, by the writer, in regard to his aim in this discussion, is very essential in order that the results may not be misunderstood. It is not intended to cover any of the ground gone over so elegantly by St. Venant, Clebsch and others. Their investigations leave nothing to be desired.
It is intended to point out considerations which, it is believed, will account for the great discrepancies existing between the results of the "common theory" and those of experiment. Those considerations apply chiefly to the conditions of stress existing between the elastic limit and rupture, to which the investigations of the authors mentioned above do not apply.
It may easily by shown that the logarithmic law found is not consistent with the equations of condition (4), (5), (6) and (7) for a body of homogeneous elasticity, but those equations do not obtain beyond the elastic limit, nor for bodies that are not homogeneous (and non-homogenity is characteristic of all bodies used by the engineer), nor indeed are they strictly true for homogeneous bodies except for indefinitely small strains. Now indefinitely small strains are by no means those which accompany the application of finite external forces or the existence of finite internal stresses.
Again the researches of M. Tresca, in particular, but also those of Prof. Thurston and others* show that molecules rearrange themselves, to a greater or less extent, when the material in which they exist is subjected to stress for a finite length of time. It is not only possible, but highly probable, that this rearrangement enables the molecules to take such positions as will give the material the greatest possible capacity of resistence.
It is submitted, therefore, that, while it is altogether probable that that condition will exist just before rupture, which, by the principle of least resistance, will subject the material to the least stress, the same law, on the further investigation of strains in either homogeneous or non-homogeneous bodies, may be found to hold in the case of such bodies in equilibrium. For that reason some approximate values for the deflection are found which may serve the purpose of (at least) a rough experimental test.
The importance of the bearing of these matters on elastic bodies, is enhanced by the fact that no law of stress whatever can exist in such bodies in equilibrium which may not be supposed to exist in a rigid body.
The arbitrary functions of integration in and are not all found, for they are not needed for the purposes of the investigation, and a search for them would cause the paper to reach far beyond its proper limits.
[*As, for instance, Eaton Hodgkinson, who, we believe, made accurate determinations in whose names are above mentioned, having turned his attention to it as early as 1824.—Eds.]
