BEAM OF EECTANGULAB CEOSS-SECTION UNOEE ANY SYSTEM OF LOAD. 81 
= _o) 
2PE 7 
cos ma X' ” «„ + . 
m- 
8ju (X' + yu,) 
1 7H 
Sin mx 
I /_1 , 1 \ 3y ® «« - /S„ . 
"r U, ^ i-^ ;— sin 7nx 
+ 4/i 
■A' i- /X yu, / 8b^ 1 m 
3y f 3X' + 4/x f _ 7X' 4 
8b [ 6fi (X' + /i) lOii (X' 
10^ (X' + /u.) f 1 
?«- 
sin ?>?x 
. (58). 
26'^E 2 
m-' 
1 2/a 
4m2Z)3 
I ^ /3);) \X + /A yti , 
T 2 HI 
" 1 +iVi+^V-’»’‘= 
‘i ! li / , my\ 
1 + “ ' - COS mx 
1 1 
1 - 
5 
__3 , r- — 7?/ /- * COS ma ,„/ 1 INloo^—/9 
2i*E 2 ■' 7 ^') + 3- + jJ -J- S cos mx 
. ^ J 13X' + IC^ \' /] « (5,_ _ 
^ 81: 110^ (V + ^) - LV (X' + m) a J T “.(59). 
CO 
Now S (a„ — /3,,) COS niac = L, where L is the difference of stress on the toji and 
bottom, in other words, the transverse load per unit length of the beam. 
a« — /3„ . 
Sin mx 
1 711 
where S is the total shear at any section. 
a» — /?« 
^ = f l^dx = f Ijilx = — 8, 
• 0 J a 
V ~ 
I 711^ 
S -'—cos mx - % cos 77ia — + | Sc/x - - M, 
where M is the bending moment at any section. 
Integrating again ; 
"to 
® Ctfi *“ yS 
^ ills ■— X z -o— cos 77ia 
1 'H J ^11-' 
V AT 
Mc/iC 
V «« - /3« s «« - /?«, 
^ -...i A -:— 
1 ■Hi* 
1 m‘ 
, cos wa; - ^ S cos wa 
1 
rx / 
J M M.dx ) dx. 
Also if Q is the transverse tensile stress at any section Q = ^ 
* «« + /3„ . 
? 2« 
Sin mx 
( Qc/ic. 
Jo 
Substituting from the above values into the expressions (58) and (59) for U and V, 
we find, remembering that — ji 
VOL, CCI.—*A. 
v + /A ' /i “ E V — — + 2/r), 
M 
