BEAM OF FECTAXGULAR CROSS-SECTION UNDER ANY SYSTEM OF LOAD. 97 
V., = - 
+ 
-\y r /_ 
n-h Jo \sinlU 
-F y If- + g 
I 
J 0 
+ 
1 2^Y// 
/u. Trh 
fJb Trh 
2W/ 1 
TT \A' + fJU 
2\V 
77 
COS ■“ cosh -7- 
0 1 ) 
*■) 
O 
'Sv? 
--) dll 
8^ 
sinlF 2;{< — 
. , 70 / 
COS 7 sinh - 
0 0 
^IL 
^ hii^ 
da 
+ -)f 
siuIF2« — 4«~ 
70J ;/// 
cos , cosh 
b h 
16?F 
ii 
smh=2«-4,^ cos|‘sml, 
20/63 
an' 
o 
y - ,u ^ 
0276"^ &3 "1“ 
da 
3 y 
1 &?63 
cZ?6 -)- Bo 
( 83 ). 
X + yU, 
where /8, Bj^, B^ are arbitrary constants. 
The expressions Ui, 'V j agree Avith those found by Boussinesq in the paper referred 
to above (‘Comptes Bendus/ vol. 114 , pp. 15 U 0 - 151 G) Ibr tlie displace]nents when h 
is made infinite. We see that U is indeterminate and V infinite at the point where 
the concentrated load acts. 
^01 couise such infinite and indeterminate dis])lacements could not occur in nature. 
\\ ith any real material, if it Avere possilMe to a]/proximate to a true knife-edge, the 
infinite stress under the knife-edge Avould at once either cause the material to break, 
or else—and this is Avhat must almost always occur in practice—reduce the parts in 
the immediate neighbourhood of the knife-edge to a jdastic condition, so that in this 
region the equations of elasticity Avoiild no longer apply. 
Hence for practical applications Ave liaAm to exclude the actual line of application 
of the load, r = 0 , and a very thin cylinder surrounding it. If Ave do this, then all 
our results will be valid for points Avhose distance from the knife-edge is at all large 
compared Avith the radius of this thin cylinder. A notable point about the results 
( 82 ) IS that Ui is indei/endent of r' and depends only upon the angular co-ordinate 
of the point considered with regard to tlie knife-edge as origin. "Hence all points 
lying on a plane thinugh tliis knife-edge receive the same horizontal displacement 
Ihe parts Bo, ^y, of the displacements are finite, one-valued, and continuous 
througljoiit the l/eam and oA^er tlie edges. 'J'hey can l/e, like tlie stresses lb, Q.„ 
expanded in senes of poweis of r, Avhicli are absolutely and uniformly comergent 
Avithin a circle of radius 46 . 
These expansions are easily seen to be the folloAvin*^'' : 
Uo=- 
+ 
Vo = - 
ixirb 
2W 
00 
'sill V(j>' 
■ V : 
77 yA ~t /6 yU. 
sin(2t/-H) (j)' 
(2r-f 1) ! 
H,„ 
1 
V 
TT A, -f- yU. j 
i/\ 2v 
sin 2v<p' 
r “ 
2(.-l 
2W//', 
IJ.TTh 0 
2W 
TT A'' -1- /i 0 
VOL. CCI.—A. 
0 VV ^ ^ '' vl 77 UWya^yt^, 
H, 
+ 
1 ^ f 2i/-H1 
-V / _ 
cos(2y -f 1)^' 
(2i. + ) ! 
00 / 
cos 2v(j)' 
'(2a)r 
(84), 
*-2y 
o 
