Deflexions of Membranes, Bars, and Plates. 355- 



x=b ; and the solution shows that whether the ends be 

 clamped or supported, or if one end be clamped and the other 

 free or supported, w is everywhere of the same sign as Z. 

 The conclusion may evidently be extended to a force variable 

 in any manner along the length of the bar, provided that it 

 be of the same sign throughout. 



But there is no need to lay stress upon the case of a 

 uniform bar, since the proposition is of more general appli- 

 cation. The first integration of (1) gives 



£( B SH> +0 ' (2) 



and ^ZdiV—0 from x = at one end to x = b, and takes 

 another constant value (Zj) from x — b to the other end at 

 x = l. A second integration now shows that ~Bd 2 iv/dx 2 is a 

 linear function of x between and b, and again a linear 

 function between b and I, the two linear functions assuming 

 the same value at x = b. Since B is everywhere positive, it 

 follows that the curvature cannot vanish more than twice in 

 the whole range from to I, ends included, unless indeed 

 it vanish everywhere over one of the parts. Tf one end be 

 supported, the curvature vanishes there. If the other end 

 also be supported, the curvature is of one sign throughout, 

 and the curve of deflexion can nowhere cross the axis. If 

 the second end be clamped, there is but one internal point 

 of inflexion, and again the axis cannot be crossed. If both 

 ends are clamped, the two points of inflexion are internal^ 

 but the axis cannot be crossed, since a crossing would involve 

 three points of inflexion. If one end be free, the curvature 

 vanishes there, and not only the curvature but also the rate 

 of change of curvature. The part of the rod from this end 

 up to the point of application of the force remains unbent 

 and one of the linear functions spoken of is zero throughout. 

 Thus the curvature never changes sign, and the axis cannot 

 be crossed. In this case equilibrium requires that the other 

 end be clamped. We conclude that in no case can there be 

 a deflexion anywhere of opposite sign to that of the force 

 applied at x = b, and the conclusion may be extended to a 

 force, however distributed, provided that it be one-signed 

 throughout. 



Leaving the problems presented by the membrane and the 

 bar, we may pass on to consider whether similar propositions 

 are applicable in the case of a flat plate, whose stiffness and 

 density may be variable from point to point. An argument 

 similar to that employed for the membrane shows that when 

 the boundary is clamped anv contraction of* it is attended 



2B2 





