Plane Elastic Systems by Louis M. Milne-Thomson

By Louis M. Milne-Thomson

In an epoch-making paper entitled "On an approximate answer for the bending of a beam of oblong cross-section lower than any method of load with targeted connection with issues of targeted or discontinuous loading", acquired by way of the Royal Society on June 12, 1902, L. N. G. FlLON brought the suggestion of what used to be in this case referred to as through LovE "general­ ized aircraft stress". within the related paper FlLO~ additionally gave the basic equations which exhibit the displacement (u, v) when it comes to the complicated variable. the 3 simple equations of the idea of KoLOsov (1909) which was once accordingly built and stronger by means of MUSKHELISHVILI (1915 and onwards) might be derived at once from Filon's equations. The derivation is indicated by means of FlLO)!E~KO-BoRODICH. even though FILO)! proceeded immediately to the true variable, traditionally he's the founding father of the trendy idea of the applying of the complicated variable to airplane elastic difficulties. the strategy used to be built independently by way of A. C. STEVEXSOX in a paper obtained by means of the Royal Society in 1940 yet which used to be now not released, for safety purposes, till 1945.

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Plane Elastic Systems

In an epoch-making paper entitled "On an approximate resolution for the bending of a beam of oblong cross-section below any process of load with precise connection with issues of focused or discontinuous loading", acquired by means of the Royal Society on June 12, 1902, L. N. G. FlLON brought the concept of what used to be thus referred to as through LovE "general­ ized airplane stress".

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Theorem (iii). ::, z) is one-yalued in D, then [f(z, z)Je= any circuit C in n. 3* ° for 36 II. The complex stresses and their properties in the isotropic case This is clearly the case for a function F (z) holomorphic in D. When is many-valued the cyclic function may vanish or not according to the circuit chosen. Thus for example, for a circle C, [i In zJc= 2 nii or 0 according as C does or does not enclose the origin. Theorem (iv). The partial derivative of the cyclic function with respect to z (or i) is equal to the cyclic function of the derivative.

40 (1) and (2). 60 hypothesis (ii) the second terms on the right-hand side vanish when A. B is replaced by the closed contour C. 60 (3). 60 (5), [zz W(z)Jc is imaginary and therefore contributes nothing to the moment. Therdore we can write (X - i't°) - (X II - i Y Ji) = -~ , If (::) (1) . e~) ["71'(:) -:: 'u:(':)]c (3) c J[ll = - Re 'Y' rf, ~~ -~:-~ ad: c Ii;; ~. 725. 13 (5) S 5 where S is the area within C. Mn= Re ¢zV Iii e = - Re 2 i I~e 2 i = - j -a~ (zV) dS 5 j z ~> d S since V is real. S Therefore J1u= -Re liz (b 1 s (6) ib 2 ) ciS.

Convention regarding sense of description of a contour \Vhen the region occupied by the material is bounded by one or more non-intersecting simple contours, we shall agree that the positive sense of description of each contour shall be that which leaves the material on the left. Thus in the case of a plate bounded externally by the contour Co and internally by contours C1 and C2 [fig. 10 (i)], the positive sense of description is counterclockwise for Co and clockwise for C1 and C2' Fig. 10 (i) \\'e shall designate by L the left-hand region occupied by the material and by R the right-hand region free of the material.

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