330 



MR. L. F. EICHAEDSON: APPEOXIMATE AEITHMETICAL SOLUTION 



It is remarkable that the changes in 3x/3s and 3x/3<? depend on the angle turned 

 through, but not upon the sharpness of the corner, as is evidenced by the absence of 

 R from (11) and (12). These equations are the conditions that the tangent planes of 

 the surface x = f( x > V) j us * ; before and just after the corner shall be parallel to one 

 another. Thus the boundary strip of x can be integrated independently of the body 

 equation ^*x = 0. 



4 - l'10. The following theorem about the total changes in 3x/3a - , 8x/8z, and x over 



any length of boundary affords a useful 

 check on the integration of equations (6) 

 and (7). Let AB be any part of the 

 boundary. Cut off in imagination a portion 

 of the solid by horizontal and vertical lines 

 through A and B meeting in D. It will be 

 assumed at first that AB does not intersect 

 AD, DB, except at A and B. Now apply 



to AD a. normal stress ZZ H and to BD a normal stress xx a , arranged in amount and 

 distribution so as to balance the forces acting on the boundary AB, together with the 

 weight of the portion ABD. Let the weight be <jpv acting in a line distant I from D, 

 and let the stresses on the real surface AB be equivalent to a force X, Z acting 

 through D, together with a couple G, The forces X, Z are to be reckoned positive 

 when they are directed out of the solid. As the body ABD is in equilibrium x> 

 3x/3.*', 3x/3z are single valued, and if there are no infinite stresses they will be 

 continuous. The changes along AB can, therefore, be obtained by integrating along 



fA ^ 



AD and DB. To balance the horizontal and vertical forces 22 d. 



Jr> 



Fig. 4. 



j-i) ^-, 



and xx dz 

 JB 



vypv 

 = 0, where v is + 1 if AB is above AD and 1 in the reverse 



condition. 



Now, substituting 



\ 



v(jpv and 



n 

 I) 



and 



xx = 



82" 



there results 



Also, since the forces on AD, DB are 



purely normal, f>xl^ z ' ls constant along DA and ?-^ff)x is constant along BL). There- 

 fore, remembering that by (11) and (12) d%/dx, 8x/?2-are continuous at the corners, 

 we see that 



and 



(13'), (14'), 



where a is the length DA. These are the required total changes in 3%Jdx and 

 Lastly, to balance the moments about D, we must have 



[A -, pD ^. 



22 (x x a ) dx + xx a (z 2 ) dz + G + 

 JD JB 



v = 0, 



