BEAM OF RECTANGULAR CROSS-SECTION UNDER ANY SYSTEM OF LOAD. 121 
W rite 
cosh u _ (1 _ 4,, + 
2 (sinh 2u + 2v') 
— e " + 6 “- 
2 (sinh 2u — 270 
= g-'' J_ ( - 1 - 477. + 
2 (sinli 2?/' + 2?/) 
= e-n . g-. 1- 1 + 4^/ + ^-3-0 
2 (sinh 2?«. — 2?7) 
and consider separately the parts of the integrals dne to the first and second terms 
of the right-liand sides of the above equations. 
We finfl, after some reductions, on writing 6 — // = iy\ y'^ (x — 1 )' = rp, 
y — y'^ y ~ 4 -. + ly = r/, {x 4- l)ly" = tan (^o, {x — 1)1 y' = tan 
sinh 2 11 
+ 
2u 
cosh 
u 
sinli 2ii 
— 
2ii 
sinh 
u 
sinh 2i( 
+ 
2u 
sinh 
u 
sin1i 2?/ 
2u 
F = - 
M^ cos 0, 
TT/'i 
Q=~ 
M'’ cos 
TT/-, 
Q wp 
sin 20J 
= ■ 0 
TT/y 
TJ’ro TTi 
+ --(P COS 2 <j^)i + - cos 203 + P, 
\ 
TTK-, 
Wy' 
irr^ TT?' 
-^2 COS 20J — COS 200 + Qo 
irr^~ 
(»«). 
where 
MY 
TT^'o^ 
sin 200 + So 
J 
) _ I (1 + 5« — 4?<“ — (1 — v) c 2") e " 77^ uo: , 7 /?/ , 
" COS ~ cos — cosh Y dll 
0 I) 0 
nil 
sinh 2u + 2u 
M j (1 + 5i/ — + (1 — 70 r-^")c " . vl . v:c . , vij , 
2» - 2„ - T 1 , 7 , 
irh 
+ 
M ,y I 7/(1 '4 4// — e '") e '' ?// ?///’ . vy . 
4 cos cos — sinh ' cD/. 
0 h h 
Jo sinh 2// + 2//. 
ivlr . 
p(l + 4// + e “'0 e " . g/ . gg 
, “Vni;2«~2“ S'll ,, sm y cosh - c/a, 
1 (1 + 3// + 47/” — (1 + 11 ) c ^'0 
jr]) 
sinh 2// + 2u 
ly v.x uy - 
cos cos -- cosh du 
h /> h 
— r"" A + 47/- + (1 + n) e~~”) />-" . It! 
( 100 ), 
■nh J 
sinh 2// — 2ii 
Sin - - sin Y sinh ~ da 
h h h 
id 
ML/ Y // (1 + 4g, _ e-27q g-„ 
vr/h Jo sinhYiYTY I t 
1(X 
+ f 
tt/F Jo 
VOL. CCL—A, 
" 7 / f” « (1 + 4« +<•-») . ul . nx . „„ . 
siiili TVTT/ s'n ^ sin J cosh (hi, 
R 
J 
