74 
MR. L. N. C4. FILON ON AN APPROXIMATE SOLUTION FOR BENDING A 
§ 7. Expressions foi' the Displacements and Stresses. 
Substituting the values of the constants found above into the equations (26)—(30), 
we obtain for the mean displacements U, V and for the mean stresses, P, Q, S the 
following values, in so far as we merely consider the terms corresponding to n = a 
positive integer ;— 
U = 
(u„ + 
1 
1 
sinli mb — — mb cosh mb 
fj, fi 
+ (P;l — V,i) 
^ + — ] cosh mb -^ mb sinh mb 
M + /U. fx 
2m (sinh 2mb + 2mb) 
= co.sh rny sin mx 
+ S 
1 
(«„ — I3„) ^ —-— co.sh mb - — mb sinh mb 
1' 1 
1 - + 
1 ^ 
1 sinh mb —^ 7nb co.sh 
+ fX 
F / 
’ i 
2m (sinli 2mb — 2mb) 
sinh?ny sin?nx 
+ S 
1 
+ 
IXn + /3« 
sinh mb 
+ 
Kn — Vn 
cosh 7nl) 
2fx 
sinh 2mh + 2m& 
2/X 
sinh 2)iih + 2mh 
^il ^71 
cosh mh 
+ 
/C« + Vn 
sinh 7nh 
2ix 
sinh 2'mb — 2mh 
2/X 
sinh 2mb — 2mb 
y sinh my sin mx 
y cosh my sin mx 
+ 
— (7« + S„) 
^ sinli mh - ^ mh cosh mb 
+ (Xn — 
A + /i 
I 1 
- Hcosh mh -^ mb sinh mb 
\\' fX fX J _ fX _ 
2m (sinh 2mb + 2mb) 
cosh my cos mx 
+ 
— (7-- — 
^ — cosh mb - '^ mb sinh mb 
-f- yU 
+ (^,i + ^n) 
1 
1 \ I 
-I-) sinh mb - mb cosh mb 
fx fx / A*"_ 
2m (sinh 2mb — 2mb) 
— sinh my cos mx 
^ , ( 7 « + sinh , (Cl — 0h) cosh mb , . , 
^ 1 — V ^ --L --—- [ u smh mu cos mx 
I I O.. OlilZi _L O.. 7. t I 
'jjb sinh 2)ah F 2mb ' 2/x sinh 2mh + 2mb 
* r -{y,i-^n) 
■^T L 
cosh mb 
sinh 2mb — 2mb 
+ 
(Cl + ^n) sinh mh 
2fx sinh 2mb,— 2mh 
y cosh 77iy cos mx (44). 
