98 
WE. L. (4. FILOX OX AX APPEOXIMATE SOLUTIOX POPt BEXDtXG A 
’where JI_i is an arbitrary constant, and the other H’s have the same meanhig as 
before. 
Idiese eijuations represent tlie effect of the finite height of the Ijeani upon the 
displacements. If in them Ave put y' = 0, </>' = 7r/2, Ave Irave the alteration in the 
displacements oA’er the upper surface due to the finite thickness. This giA^es us, 
retaining only the leading terms, 
2W / 1 
(V.,) , = 0 =-U' : - 
^ 77 \A. + /U- yU,/ 
] \ H, _ 
4W 
■1 ’ Ir 
, ( 0598) ,2 , 
(U.),-=o = 
2A\ 
1 
TT \V + ya + y. ) h 
ttE 
1'93365\ 
7rE5 
■47SW 
giA’ing ; 
a d(»A\n\A’ard curvature at the point of discontinuous hjad etjund to 
a horizontal stretch 
1^34W 
7rE5 
The effect of the fiinte thickness appears therefore to 
he to stiffen the heani and to decrease its curvature under the load. 
§ 20. Expansions about Other Points, Expansion about the Origin. 
The exi)resslons (71) and (72) are capable of being expanded in many other Avays. 
Considering only expansions in poAvers of the radius A'ector from a giA eu point, a\ e 
may write in U, V, P, Q, S : x’ = X + p sin 6, y — E p cos B, and aa e shall 
obtain an expansion AA'hich is \arlid for all points Avhicli are contained betAxeen 
7 / = + 6, and A\ hich lie inside a circle Avith centre (X, \) passing through the 
point (0, + b). The coefficients of p" cos rid, p" sin nd, &c., AA'ill be integrals 
containing X, Y. 
The only expansions Avorth considering are those about the origin and those about 
file point (0, — b), AA'hich is vertically beloAv the load. 
The expansions about the origin are deduced immediately from (71) and (7_). 
They are 
U 
_ ^ Yb/ ^ /py siiip-p -p, _ IP ^ I P’ 
fX 'lTTh~^\hl v\ 277 1 \llj v\ ' + jX 
V = 
1 Y'// P /rV' cos 
CT 
fx 'lirh 0 
P = 
- V 2 
V . 
AV X / r 
V 
'izh 
-77 1 
(.'US 
V . 
r p cos i'(f) J 1 
A' 4- u 
V ■■ 
) 
(85), 
w-w.)-) 
Q = 
AV X /y. Y cosvc}, Wy^/pVeo^ 
AA" X 
s = 
irh 1 \ I 
r V sin vcj) AA"v S / r 5" sin „ 
X (T ) ::. ■ r+i 
V 1 
775'- 1 \ b 
(8G), 
V . 
J 
