122 
MR. L. X. (4. FILOX OX AX APPROXT^^IATE SOLUTIOX FOR BFXDIXO A 
W r n (1 - 4/6 + c-~'‘) c-“ V-l . y-j: • , V-V 7^, 
ft = — —^--cos , sm — smh - du 
''-^•'1 7 1 .-.^,.. 1 -. O7 . I Oi / /-. n f ) 
+ 
ttI Jo 
Av r 
irh J 0 
-1 -I r ,■* CO 
^v// 
ttM Jo 
sinh 2u + 2w 
1 4- 4 » + c- 2 «) e~" . vl ux , vy j 
2 T - 1 — sin - cos — cosh 4 du 
sinh 2ii — '2u '> '' 
(1 + 4)6 - e-“'0 c-’' id . ux , vy , 
^ - - - - -cos -r Sin 4- cosh — du 
sinh 2a + 2/' h n // 
(1 + 4 a + c-^") C-" . vl vx . 1 vy 7^^ 
4_ ' 1 — --- sm : cos — smh ~ du 
' ttM Jn sinh 'Iv — '2a h h h 
AV// ) - ic 
'0 
. (100). 
Po, Q., S.7 are finite and continuous all over the beam. They may be exj^anded in 
powers of r about the origin, the series iDeing convergent inside a circle of radius 
y/i (36)y so that the points of concentrated loading are included. The parts of 
P, Q, S which become infinite at the points where the load acts are of the same 
form as if the lieam were an infinite plate. 
§29. Series in Powers of r. 
We may here quote the expressions for Po, Qa, So in powers of r. They are 
Po = 
_ w ” j r cos 2v(f) I" 7(> (1 + 5u - 4a~ - (1 -'v)e -») c ” 
ttI/ 
(2v) 
sinh 2 a + 2 a 
uJ 
AAr X ! y yr f-iii 2 z7(^ [■“ vP (1 + .5a - 4?d + (1 - »)f "'0^ " 
’T^li \ \ l' J ! 
sinh 2 a — 2 a 
AAh/ ^ fr'y"+'^ c os 2^r^ cf) U"‘'+" (1 + 4m — c " 
ulfi o\lij (2i' + l)! . 
sinh 2 a + 2 a 
v.I 
COS 4 du 
0 
V\ il * / r sin 2v + i(j) p zr‘'+-(l + 4 ?a + e ~") c rl 
^TrfrUXh) (2a+ 1)! Jo sinh 2a - 2a, h 
AV ” / r ' P cos 2 v 4 > p v ~'’ (1 + 3a + 4a- — ( 1 + a) c ~") c ul 
.'0 sinh 2a + 2a ‘ h 
AV ^ sin i 
2a<|> 1 ” a-^>'(l + 3a + 4/d + (1 + //) e--") c-- " y/ 
^h-\hl (2zhrJo sinh 2a-2a h 
AV// / /• cos 2^^^ (/) p a- '^+-(l + 4// - c" -") V” ul 
irlr n \ h 
(2a + 1) ! Jo shill 2a + 2/(- 
COS - du 
0 
AV//^ / 9 ’ \~‘ + ^ sin 2a + l (p p //“'''"(I + 4// + c ~'')c " • 
+ -- t[— 
Trh" 0 ^ 
(2a + 1): 
sinh 2v — 2a 
sm -7 du, 
b 
