BEAM OF EECTANGULAR CROSS-SECTION UNDER ANY SYSTEM OF LOAD. 91 
T) , ^ 2Y’' f” sinh u ux , uy , 
r H- — r ■ , ^ -— cos -7 cosh dv. 
7r/> J 0 siiih 2?< + 2?? 1) 
r 
Trh J, 
0 sinh 2 ?< + 2 ?? 
cosh u 
m 
irh Jo [sinh 2n — 2?6 
tiK . , uy oy 
cos Sinh --- - 
dii 
- O = \ 
7rh Jo 
1 1 '>'v -1 -1 '?o/ 
V cosh u cosh - — •- smli u sinh 
sinh 2n + 2u 
VAj 
~h 
1) vx , 
-- cos -- dv 
b 
u sinh ?/ sinh ~ cosh n cosh 
sinh 2\i — 2ii 
h i(x , 
— COS — + 
-^sJL 
^ hv} 
y (73)- 
dv 
P — Q and S give the lines of principal stress and the principal stress-difference, 
P + Q gives the compression at the jDoint considered. 
If in the values (73) and in S we write y = b — u' m that rve are referring our 
co-ordinates x, y' to the point C (fig. i.) wliere the concentrated load is applied, as 
origin, we find, on re-arranging the terms. 
PJ_ 0 - 2 fVr-r nx 
c + m =-r- e '' Gc>s---dic 
Tfb J 0 u 
_ _ f 
irh ]( 
I ri + 2 a + . 
J 0 t ^ sinh 2 i 6 — 
u 
sinli 2u — 2u sinh 2u -f 2 « 
vx vy' o I 
cos —- cosh — tv; !■ dv 
0 h air 
’Al, 
1 -f- 2ii — c“-“‘ 
sinh 2 » -f 2%i 
vx . VAf l> ?/■ 
I T - 4„1 T 
dv. 
vy' 
^ 2Wy'r - _ 
t — = —dr '' cos - du 
irb'' J 0 
ux 
~h 
— f 
irh j 
V 
V. 
0 L — 2ii sinh 2u -f 2v 
/ .00 , - r. 2u + e--^ 1 -f 2v - 
vx vy' c! , , 
COS 7 cosh — r , 1 d'U 
h h 4 ?d 
^Yi/ r r.i n 
~l:r Jot 2" [si 
irJr 
2\Yy' 
s = 
2 '' V r 
“ Jo 
fYi/' 
sinh 2u — 2 ?i sinh 2u -4 2v 
vx vx 
vx vy .! , 
cos -7 cosh - i- dn 
li b avi- 
irW 
_sin]i 2 ?i — 2u sinli 2u -f 2u 
vy' 
'iix , ^ iiy , 
cos T smh -7' dAi. 
b b 
— ~ ~ 'lOX 
'tie '' sin — dv 
0 b 
+ d f 
mb 
f 
u 
u 
Jo 
sinli 2v - 
- 2v sinh 2u 4 - 2%i 
UX 
sin sinh -f- dv 
b b 
_ W r_v 
L 2 
1 + 2i( -j- c--“ 1 -I- 2?t — 
ttY Jo 2 Lsinli 2 ?i — 2u sinh 2u + 2 ?( 
. vx . , vy , 
sm ■ 7 smh -Y- dv 
h h 
70~ 
U“ 
_sinli 2u — 2u sinh 2u -f 2 ii 
N 2 
l(X 
sin ‘ 7 cosh - 7 - du. 
b b 
