BEAM OF RECTANGULAK CROSS-SECTION UNDER ANY SYSTEM OF LOAD. 85 
/ = 
4&'5E 7 
cos UTT W 
a 
COS 
nirl 
— 1 
a 
+ 
; w 
7 a 
nirl , \ 1 
COS -— 1 
a 
X' fj. 
, 1\ , mh 
i— cosh mh -j-siiih mh 
I 
m. 
Now the first term can be evaluated. It is 
sink 2mh — 2mh 
3 \Y 
nirl V , 
cos--^--l). (G2). 
16 ah'^ E' 
We have therefore 
_ 3Wd 7 
— IG'EWWT^^iTr 
I nirh 1 /' nmh iiirh . nirh 
wT \ ^' t cosh “I f cosh “j siiih 
VV \A 4- a a \ a a a 
/ 2nirh ‘Imrh \ 
sinh--- 
(0 a / 
' n7rr\^ 
1 — COS — . 
a / 
Now let ns remove the ends to infinity, that is, make a very large. This will 
transform the N above into a definite integral. It is easily seen that the term under 
the S remains finite and continuous when n is made zero ; we may therefore take 
our limits from 0 to co. We then obtain, putting mrhla — v, irhla = du : 
/ = 
1 \ ii, 
-VI-. I ^ , 4“ cosh +— siiih xo 
\v \ A. +yu. fji / pu 
sinh 2u — 2u 
4 u 
—cosh u + — siiili u , 
Tj //. ! 
'o -tt 
COS 
'?d\~ (lic 
or writing I/2h — 
2\Y 
TT 
1^ 
ul Y^' dn 
sinh 2u — 2 il 2h) u 
2XY ( I \‘t 
“ TT \2^ 
4id cosh u sinh u 
E 
sinh 2ii — 2ii 
\ ^ ^0 
(63). 
Now (sin wXq)’- is always < 1, so that 
2W / / X-i 
y < -w 
TT 
2h 
4 sinh nh 
,, %(? cosh ii +- 
fj, 
sinh 2u — 2ii 
dll, 
and f tends to become ecpial to the right-hand side of the last written inecjuality 
if Ij'Zh becomes small, that is, if we make our supports close up. 
?d sinh % dll, , . 
— - - when calculated by quadratures 
rp, . , 1 f cosh u die , f 
I he integrals . , ^ and 
° Jo sinh 2ii — 2u j 
0 sinh 2u 
come out to be equal to 7'22 and 24*82 respectively. 
We have therefore 
/ < 
2W / I /28'9 
2h 
24-8 
E ^ 
. . . (64). 
Now if J-Q be the Euler-Bernoulli deflection, that is, the deflection calculated in 
the usual way by taking the curvature proportional to the bending moment and 
fixing, so that the elastic line is horizontal at the origin, 
YiP 
fo = 
(65). 
Comparing (64) and (65) we see that the true deflection will certainly be less than 
