. A treatise on the mathematical theory of elasticity . s it isnearly horizontal. * See for example the treatises of Eankine and Grashof quoted in the Introduction, footnotes 94and 95, and those of Ewing, Bach and Foppl quoted in the footnote on p. 110. 352 CRITICISMS OF CERTAIN METHODS [CH. XV (6) In the extension of this method to sections which ai-e not rectangular it isrecognized* that the component Y^ of shearing stress must exist as well as X^. The caseselected for discussion is that in which the cross-section is symmetrical with respect to avertical axis. The following assumptions are m
. A treatise on the mathematical theory of elasticity . s it isnearly horizontal. * See for example the treatises of Eankine and Grashof quoted in the Introduction, footnotes 94and 95, and those of Ewing, Bach and Foppl quoted in the footnote on p. 110. 352 CRITICISMS OF CERTAIN METHODS [CH. XV (6) In the extension of this method to sections which ai-e not rectangular it isrecognized* that the component Y^ of shearing stress must exist as well as X^. The caseselected for discussion is that in which the cross-section is symmetrical with respect to avertical axis. The following assumptions are made :— (i) X, is independent of y, (ii) the resultants of JT, andF, at all points P which have a given a: meet in a point onthe axis of x. To satisfy the boundary-condition (3) thispoint must be that marked T in Fig. 31, viz. the pointwhere the tangent at P to the bounding curve of the sectionmeets this axis. To express the assumption (ii) analytically, let rj be theordinate (iVP) of P and i/ that of P, then .(37) Equation (2) then becomes ex r)cx 1. and the solution which makes X^ vanish at the highest point {x= —a) is W prtX^= --^ / Xrjdx, and it is easy to see that this solution also makes X^vanish at the lowest point. The stress-system obtained by these assumptions is expressed by the equations Y V y c Y ^ l ^ ^^ Y ^y^l [ ^ ?^ Wx{l-Z) X^= r, = Xy=0, ^Y,= -^j _^a;r,dx, Y,= -—^j^^^;dx, Z,= ^^; (38) it satisfies the equations of equilibrium and the boundary-condition, and it gives the rightvalue W for the resultant of the tangential tractions on the section. But, in general, it isnot a possible stress-system, for the same reason as in the case of the rectangle, viz. theconditions of compatibility of strain-components cannot be satisfied. (c) These conditions may be shown easily to lead to the following equation :— i{JSf_/^^4 = i^ (39) which determines ;; as a function of x, and therewith determines those forms of section forwhich the stress-system (38) is a possible on
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