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

By A. E. H. LOVE

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Sophie Germain, and her work Institut was published in She assumed that the sum of the 18211 the plate when bent plays the same part in the theory of plates as the curvature of the elastic centralline does in the theory of rods, and she proposed to regard the curvatures principal work done of the of in bending as proportional to the integral of the square of the principal curvatures taken over the surface. sum From this assumption and the principle of virtual work she deduced the equation of flexural vibration in the form now generally admitted.

Angles y8 and y with the horizontal. The equation of moments of moments 33 any part BP of BQ is Ms + B^x = 0. (35), any part GP' of CQ is -^— Mc-^G^ = 0. a^,) in td(y-xtsiny) GQ When the sum 33 (6 33 ( a; =6 in BQ, and x = of the dyjdx^ tan yS - tany8+ c tan y) tan 7) is zero. (39). (40). THEOREM OF THREE MOMENTS. 223] Now in B AB of length the span 89 a preceding 5 measuring x from we have and integrating (41), we and = 0. ,. (41), (42); ^ when tan + «tan^) = iif£^2-i(ilf^-if^)^/a when x = a, = 0, (43), 33(2/ and since y= 33tan^ = ^a(2if^ + if^) In a?

Are the points of support, Ao being simply supported, and that for any span Ar Ar+i the curve of the bending moment when Ar and Ar+i are simply supported has been drawn. , i' acting through the point ^/nearer to A^, cj^^, and (fy^' through the points g^, and g^ and so on (see fig. 30). Let F^, F^... be the forces that act through (ti, 0^... where the line A^, A^... meets the verticals through the centroids of the areas of the curves of the bending moments for the various spans when their extremities are simply supported.