92 
MR. L. N. G. FILOX OX AX APPROXIMATE SOLUTIOX FOR BEXDIXG A 
The leading integrals in each case can be evaluated. 
If we write x = r' sin y' — r' cos (j)', so that r' is the distance of the point 
considered from the ])oint of application of the concentrated load and (f)' is the angle 
which ')•' makes with the vertical, then : 
h cos </)' 
ux 
X) vy' 
e '' cos -7- d u = 
0 '' 'I" 
poo v.y 
ux , cos 2(/)' 
ue '' cos XT du = 
1 0 r 
ux 
and we have 
f oo vi/ 
ue '' sin '7 du = 
0 
Ir sin 2(f)' 
2^Y 
P -f Q = — ' y cos (f)' — 
HW 
irr 
V“ 
- COS \ cosh -7- — 77^7 [du 
nt/ 
ttI) } a sinld 2io — -iir 
16//- 
u 
.,,2 I _ 1 i 
" « UX , Ud ?, y' 
il-h . , njj »') 7 7 
siulr 2//. - 4/d T ~ Toid I ' 
8W 
«“ 
irh Jo siuld 2%i — 4?6- 
ux , Ulj 
COS — cosh y- 
3 
16/d 
du 
ux 
-( 
h 
uy o 
1) 16 //- 
c/ W//' . , 4tVf=“ ^d . nx . , uy' , 
b —7 Sin 2 (f> + -, ’o sin -- sinh du 
irr - ' TTO Jo Sinn- 2 n — 4 /t~ 0 0 
4W// 
, ' u~ u u 
/ i -t y + o - o e 
2 o 8 . lit: . uy , 
70 1 — • • I o .0 -y;—sm -7 smh du 
170- Jo Sinn-2?/— 4?f~ h h 
4’\Y//' r” . ux . uy' j 
+ —hr .y v.-, -ry sm — cosh : du 
ttP joSinld2// — 
4/d 
(74). 
(75). 
(76). 
The expressions for the stresses therefore consist of two parts, namely, the 
integrated paiTs 
( 77 ), 
S 7 = 
and the parts still in the form of integrals, v'hich we may call Po, Q.^, So. 
w 
\Yy' 
2 AA" 
.)-y'\ 
- 7 cos 
•nr 
+ 
,0 COS 
ir r ^ 
2 (f)' = 
IT 
AV 
AAG/' 
2 ^Y 
—, COS 
irr 
cji' - 
.,7 COS 
irr ~ 
2f = 
IT 
1 
—^ sin 2 fh 
nr - 
rr; 
IT 
/o 
y/" 
^.'4 ) 
