BEA.M OF KECTANGULAli CliOSS-SECTlUN UNDEii ANY SYSTEM OF l.(JAj). 
18 
Bcisy to S 06 tliat I — j / (w) siii chi tends to zero cis ^ tends to intniity. F* 
integTating by parts 
I 
— y COS 11^ fill) 
-f- ^ COS a I f {it) dll 
1 
= - cos if ffu) du. 
J8iit 
cos if f 
• AT 
^ I \ < a finite quantity M; lienee I <- 
•o' . ^ ^ 
and therefore tends to zero as ^ tends to infinity. 
Hence, when a; is large, V reduces to tlie second integral. The h 
evaluated, and it conies to 
can 1 le 
■i T 
3 '2-*^ 
1 O 
+ -a: - 
L /9 
A' + yU. yU, Ir 160 \/U, 
for a: > 0 and 
1 , \\x^ , F /9 1 \ 
~ U' 4- “I- ) t ^ “1“ irn ' ~ ;- 
•)_ A + 1 IGO'/i X + yU,/ 
1 he lirst terms correspond to tlie bending due to the shears at the ends. 
We should therefore try to make k!x — 
L /9 
of X. 
160 \yU. X' + fl/ 
= 0 for all large' values 
o 
Ibis is obviously impossible. But iVx being eventually the most inqiortant term, 
the condition is approximately fulfilled by taking A'.— 0, This determines U and 
\. We see that the effect of tlie i.solated shear L is to defect the central line of the 
lieam through the distance g x ) away from its line of action. 
Putting A'= 0 in equations (105) they give us U and V. Integral expressions 
for the stresses are obtained in like manner. They are 
P = - 
1. 2 cosh u — a siiih n , in) . 
7 --cosh ; sm — du 
ttCJo siiih + 2a h h 
L p 2 siiih u — a cosh e 
7rl/J(, siiili 2a — 2a 
L p //.// cosh u 
irh J 0 5 sill 
E f"” a// si nil a 
• 1 "H • / 
sinli — sm , dv 
0 h 
. T aa . ax , 
, siiih : sill --da. 
w + 2 a h 1) 
g// _ 
ttAJo h siiih 2// 
cosh /sin A (/a. 
(lOli) 
