By A. E. H. LOVE
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Extra info for A TREATISE ON THE MATHEMATICAL THEORY OF ELASTICITY
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. The exceptions to which we have alluded produced are more apparent than real. Kirchhoff s proof depended on the variation of the energy, and he considered only first variations. In the cases we have mentioned a small displacement really changes the character of the surface-tractions, and it becomes Take for example the necessary to consider second variations.
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.
The bending moments at the supports, ... F^, Fg... the pressures on the supports, Aq, A-i^ the shearing forces estimated upwards to left and let ^ and indefinitely near to it, ^o, -Si similar quantities for right of and so on. o + -Aj, F^ = 5o + B^ and so on. B Let w be the load per unit length of the beam. BoA aB BA BG measure y downwards and x to the left from B, measure y downwards and x to the right from B. Considering the equilibrium of any part BP of BA, and taking moments about P, we have In in ^%-^B + B,x-\wx' = 0.
A TREATISE ON THE MATHEMATICAL THEORY OF ELASTICITY by A. E. H. LOVE