492 X- STATICS OF DEFORMABLE BODIES. 



IV. ^ 4= 0. 



129) = -^ 



; = ^ 

 where Fj is defined by 



cos ( w *) + ( 2 + ^) *y cos () 



If the cross -section is symmetrical with respect to both axes 

 of X and Y, evidently the boundary condition is satisfied by a func- 



3V 

 tion Fj_ evm in y, and o<i^ in x, consequently -^ is odd ; and vanishes 



at the origin. 



The line of centers is deformed into a plane curve (since v 

 vanishes with x and y), having the equation 



a cubical parabola. The strain is called a non- uniform flexure. The 

 stresses are 



r; = &,- (2 



Z= / f Z l dS=Eb l z I I xdxdy = Q, 



T=J'fY i dS= tl 'b\- (2 + ^JJxydxdy +ff^ dydx] - 0, 



the latter integral vanishing because V(y) = V( y), and the section 

 is symmetrical. 



132) x== 



= ^ - 7, + - l I x 



the last integral taking the above form since V^(x) = FI( x). 



