BEAM OF RECTANC41TLAR CROSS-SECTION UNDER ANY SYSTEM OF LOAD. 101 
§ 21. Expansions ahout the Point (0, — h). 
It appears of some interest to give the values of the displacements and stresses 
about the point (0, — 6), that is, the point of the lower l)onndary of the Ijeam whieli 
is vertically under the load. 
The integral expressions (71), (72), (73) transform as follows, if we write 
y = y 
u 
-lx 
1 AY//" p J 2)1 cosli 2ii + sink 2a . V ' 
2'irh J 0 1 
+ 
1 AY//" 
sink- 2a — 4«- 
2u sink 2a 
sm cosh ^ f '--j (In 
0 I) *' ‘ 
A/r- J 
. v-v. .•,)()! 7 
0 7 1 -TOO i o -y- du 
Ijb IttI) j 0 sinli- 2u — 4))- h h 
1 
1 
Jtt \x + ft fjb 
2 sink 2a . ■)i:c , v)i" „ x ’ 
, 0 .o -sin - cosh -A 
sink- 2a — 4:a ' h b ^ ha- 
du 
1 
AA^ 
27 T a' -k /a 
2 cosk 2;^ + ?; ^ sink 27; . )ix . iiii" c, .r)/' 
- sm V smh ~ — -f- - 
sink- 2)' — A)i~ 
h 
/r/r 
(ht , 
V=: - 
4 - 
1 AY//" r” r 2a sinii 2a v,x . ?///" k , 
V 27r/7 J 0 1 sinid 2a - 4/4 ~ h ‘ h ~ 4?d ^ 
1 AAO/" r'" f 2/'cosk 2/7 + sinli 2/^ ux . idi" ?/" , 
2^1,1” I""’’' 
AA" 
■'> 0 
1 
1 
^ 12 cosk 2/7 + 77 1 sink 2/7 vx Vif 
.7 w/, +~ i-- T-. - cos. cosh-'-- 
-TT \\ + jj, + / .' 0 L Sink- 2/6 —1/7- h b 
i'> 
.) 
4//1 
o ./y.i 
. , // - 
8/7.3 
AY / 1 
, , ... \ f 2 sinli 2/7 a.r . 77//" 3 ?/" i , 
d--r— u, '■ i -TV cos -sink V I n77 
277 ' X + /Li/Jo [sink-2// —1//- b b 4/'- // 
d n 
P + Q = “ 'I 
2/7 sink 2/7. 
2Aa; 
irb j 0 1 sinli3 2/7 
vx 
lid T 
77// ' 
b 
4//3 
du 
2AA^ p"' 12/7 cosk 2/7. + sink 2/7 ax . , ////" 3 ;/"] 
'X ]. 1'"/ m ,= 2« - """ / 'X - '1, \ 
P - (/ 
X r 
4//' 
cos - cosh 
— - ^ 7 
4/73 
d)t 
2AAV''I” 27/4 sinli 2/7 )tx . , 77 ?/" , 
-bX J . /iulP 2„ _ 4»= T /; 
