BEAM OF EECTANGULAR CROSS-SECTION UNDER ANY SYSTEM OF LOAD. 
79 
PART 11. 
Discussion of the General Solution when the Forces on the Beam are 
Purely Normal and are Symmetrical about x = 0. 
§ 10. Expressio^is for the Stresses and Displacements, 
If the foices aie purely normal, and if the solution is to he even in cr, then the 
y, 8, f 6, K, V terms disappear. 
Fuithei, Y^e have the additional condition that, over the ends x — d:: n, T — 0, 
S 0, M 0, by introducing suitable terms of the form (52) \ve can satisfy this 
last condition, and we finally obtain 
TT _ - 
+ s 
+ S 
2m 7 
cos ma 
vd 
+ AO \p' + 
2ra 
siiih ml) 
ml) cosh mh 
sinli 2mh + 2mh 
cosh my sin mx 
2m 
I ^ (“« + /3,i) y sinh mh sinli my sin rm 
I ^ o 
1 2y, 
sinli 2riih — 2mh 
' sinli my sir 
sinli 2ml) + 2mh 
— sinh my sin mx 
3 I 1 mh coi 
sinh 2inh — 2mh 
I ^ (“« ~ dn) y cosh mh cosh my sin mx 
“r ^ '- 
2/1, 
Y = 
«o.y , - 
E 21 m 
o / -tr — rjip 
^ p X («„ - A) 
/ 1 1 
COS ma 
7)1- 
+ s 
1 
'«« + A(\ 
lU' + /i ■ 
, / 
J 
7' 1 
'«« — /3„\ 1 
[\E + y 
2m j 
+ B 
1 
+ 
sinh 2mh + 2)nh 
1 \ . . 1 
n 
cosh mh + — 7nb sinh mh 
n 
sinh 2mb — 2mh 
^ fr. 
sinh my cos mx 
cosh my cos mx 
_ y /y cosh my cos mx _ - m,, - ^A y cosh 'mh sinh my cos mx 
X 2J sinh 2mh + 2mh 7 I 2/^ ) sinh 2mh - 2))ih 
J 
(5G), 
where B is an arbitrary constant to be determined from some condition of fixing. It 
merely corresponds to a total vertical displacement of the beam. 
V = 
'-yi V / /D \ cos ma. , ” 
- f («« - A,) ^,,2 + ^ 
m- 
, ^ ^ sinh 'mb — 7nh cosh mh , 
+ "si„h imh + 2»,S ^ 
_L V / Q \ cC'‘’h mh — mb sinh mh . 
+ 7 - P'd ^ Yhil72»i/;^L.r' my cos mx 
sinh 2'mb — 2i«& 
inh 7)ih sinh 7)iy ( 
sinh 2m5 + 2'mh 
>( 57 ), 
4 - Y ((^ 4 _ P ) ‘^yyfyfpdpinhm)^ -P Y / 4 i cosh cosh cos 
t ' cn T» 1*« O/i-v^ 7-> 1 ~ r )l J • 1 "i 7 -V 7 
1 sluli 2mh — 2riih 
