By A. E. H. LOVE

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**Extra resources for A TREATISE ON THE MATHEMATICAL THEORY OF ELASTICITY**

**Sample text**

19 given in the Natural Philosophy of Lord Kelvin and Professor Tait. These authors make little use of the theory of Elasticity, but deduce the values of the stress-couples by a method similar to that which they employed in the case of thin rods. More recent expositions of the same theory have been given by Saint- Venant^, and by Mr Basset^ and Prof Lamb^ The two latter were led thereto by the difficulty in the subject of thin shells to be referred to presently. We have already noticed that the boundary-conditions obtained by KirchhofF differ from those found by Poisson.

In problems of the kind we are now entering upon the fact G is proportional to d^y/dx'^, and the ordinary principles that of Statics, are together sufficient to determine the form of the strained elastic central-line, and the pressures on the supports can be deduced. (Cf Since there (1) is i. art. ) no applied couple M, the third of equations becomes ax where dx line is written for ds since the strained elastic central- very nearly coincides with its unstrained position. This equation gives the shearing force at any point.

22. We have at once the equation of moments ax^ (8), BENDING OF RODS IN ONE PLANE. 32 [218 with the terminal conditions that y and dyjdx vanish when x and when x = L We have on integrating %= + ^^^ if - ^ Fo? 4- -^wP = 0, where iilfa;2-jFo^ Fo and as {x' + \l) (9), M = ^wl\ from which = before, referring to the for X, we find (10); \wl middle point as origin and writing ^y=iiw(\P-xy and the central deflexion is = ^w^V^ or ^ of (11), what it would be if the ends were simply supported. The reader will find it easy to prove in like manner that for a beam of length I supported at its middle point the deflexion y at a distance x from the middle point is given by the equation = ^^wx"" (3/2 - Ux + ^x% SBy so that the terminal deflexion is ji^w;^/33 or f of that at the middle point of the same beam when its ends are supported.