. Un trattato sulla teoria matematica dell'elasticità. Lacement mustaffead equations (49) dell'articolo 97. Se prendiamo u^ e UG proporzionali a cos n^, e^ per essere proporzionali a sian(/>, possiamo mostrare che* ^^cosMj (^ ()i + cos 6) tan +B (k- cos 6)cot ^l, sin nd) („, 6 , 6]S^r ^ C tan -J) cot -|, Coswdi f X-l-2u j-^A ^^ 6 „ ,„fll «r= ^J- ! ^ +etan-+i)cot5L »• i /i cos»0 2 2J dove A, B, C, D sono costanti arbitrarie; e poi possiamo mostrare che «e=^^i-- H-^smfl-T3( -l+cosfl (Ctanjr+i>cot5) r sm fl ( 2/x rffl cos ?i(^/ 2 2/ + (?tan| + fi^cot^|,„,=^L^M J^ ,„,g/(.tan»|-iJcot-f)-(

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. A treatise on the mathematical theory of elasticity . lacement mustsatisfy equations (49) of Article 97. If we take u^ and Ug to be proportional to cos n^, and^ to be proportional to sian(/>, we may show that* ^^cosMj (^ ()i + cos 6) tan ~+B (k- cos 6)cot ^l, sin nd) („, 6 _ , 6]S^r ^ C tan ~-J) cot -|, coswdi f X-l-2u j-^A ^^ 6 „ , „fll «r= ^J-_!_^ ~+etan-+i)cot5L »• I /I cos»0 2 2J where A, B, C, D are arbitrary constants; and then we may show that «e=—^^i--H-^smfl-T3( -l+cosfl (Ctanjr+i>cot5) r sm fl ( 2/x rffl cos ?i(^/ 2 2/ + (?tan| + fi^cot^|, „, =^L^M J^_, „, g/(.tan»|-iJcot-f)-(?tan-§ + ^cota, where G and fl^ are arbitrary constants. In the particular cases where «=0 or 1 some ofthe solutions require independent investigation. These cases include the first type of simplesolutions for any direction of the applied force, the second type of simple solutions, and thesolutions arrived at in Article 132, examples (d), (e). We give the expressions for thedisplacements and stress-components in a series of cases.. * J. H. Miohell, London Math. Soc. Proc., vol. 32 (1900), p. 23. 200 FORCE APPLIED AT [CH. VIII (a) The first type of simple solutions, corresponding with a force F parallel to theaxis of z, is expressed by the equations F cos 5 X + 3/i F ^6 the stress-components are expressed by the equations -— 3X-|-4u -^ cosfl S2 r> u F cose?=-^+2)7 4^-^ ^^=^ = XT2;:4^-^ The meridian planes (0 = const.) are principal planes of stress; and the lines of principalstress, which are in any meridian plane, make with the radius vector at any point angles ^determined by the equation tan 2^|^ = - {2/i/(3X + 5/i)} tan e. These lines have been traced by Michell, for the case where X = fi, with the result shown inFig. 16, in which the central point is the point of application of the force. O) When the line of action of the force F is parallel to the axis ofx, the displacementis expressed by the equations _ F sin5cos<^ _ + 3fi. F cos5cos0 _ X + 3

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• bookcentury1900
• bookdecade1920
• bookpublishercambr
• bookyear1920