## A TREATISE ON THE MATHEMATICAL THEORY OF ELASTICITY by A. E. H. LOVE

By A. E. H. LOVE

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The T along the tangent force can be resolved into two components : to the strained elastic central-line at P, and N along the normal same line drawn inwards towards the centre of curvature. The axis of the couple is perpendicular to the plane of flexure. The couple will be denoted by G, and it will be called the flexural The forces T and JSf will be called couple or bending moment at P. to the respectively the tension, and the shearing force at P. application of the general theory of Elasticity it can is proportional to the curvature By an G be shewn that the flexural couple of the strained elastic central-line at P, so that if p be the radius of curvature of this line, G=^lp, a constant depending on the nature of the material and the form of the section.

Draw from A and This B may vertical Fig. 29. A and proportional to the bending moments at the are whose ordinates B, and draw through A, B the curve is isolated when this span bending moments at the points of AA' and BB' lines AB and simply supported at A and B and is under the given distribution of load. Let the verticals through the centroids of the triangles AA'B and BA'F meet AB in g and g', they are the vertical trisectors of the line AB, and let the vertical through the centroid of the curve of the bending moment when A and B are simply supported meet AB in G.

1884. I'^lastique et I'une de HISTORICAL INTRODUCTIOX. been known, one involving or 23 and the other involving simple and it was concluded that in flexure, torsion, compression simple such cases whenever flexure is possible it will take place but there existed no theory for determining a priori when two modes of equilibrium are possible, or why flexure should take place whenever it is a characteristic of one possible mode. ; These defects were remedied by Mr G. H. Bryan \ In regard first point it had been shewn by KirchhofF that when a body to the is held in equilibrium by given surface-tractions the state of strain is unique.