90 
BEAM OF EECTANC4ULAE CEOSS-SECTION UNDEE ANY SYSTEAI OF LOAD, 
where y ■= r cos (fy, x = r sin (fy, 
u cosh u 
sinh 2t/ — 2u 4?t- 
dti, 
TTt _ r” cosh-?«■ 7 / ^ 
! 0 sinh 2u — 2u 
sinh It 
0 sinh 2u + 2u 
G-. = f .i 
2v + 1 
cosh u 
Jo sinh 2it + 2)f. 
(In {v ^ 0), 
?r‘' sinh u 
sinh 2u — 2u 
dll {v > 0), 
G(i = a constant to be adjusted from the fixing conditions. The series in (85) and 
(86) are al)soliitely and uniformly convergent inside a circle centre the origin and 
radius 5. 
The first few coefficients are given by 
Fo = 
•527 
Gi = -918 
Fi- 
•438 
G. = 2-818 
F, = 
1-740 
G. = 5-750 
II 
C 7 > 
7-224 
Gj, = 24-824, 
where the integrals have been obtained a])proximatelv by (juadratnres. 
Retaining in the expressions (85), (86) only the most im 23 ortant terms, we find 
for the disjilacements of points on the .'C-axis : 
AY V /1 
2it h \ fjL, 
•918 
which is positive witli x. 
We have therefore a liorizontal stretch equal to 1 ^44 
+ /X 
•527 ), 
^08' 
For uni-constant isotropy E = 5/x/2, aiid the stretch is - 
2&E 
AY /1-503' 
TT 
, or about one 
half the stretch due to the load W acting horizontally along the length of the l;)eam, 
so as to produce a tension W/26. 
Q- -1 1 A- . ''F a;2 1 /4iq , Cb\ i • . 
cnmiiariy _ Ltq + — — - - this gives a curvature upwards 
Lit 
. , AY /1-75?. , 2-818\ . 
equal to ^ l to tlie curvature that would be produced liy a pure 
1 AA77 /' E \ 
couple ^ (h753 + - 2-818j, or (putting E = 5p,/2) by a couple Wb X (’5622). 
The stresses at points along tlie .r-axls are 
o 2 
