Solutions for certain rectangular slabs continuous over flexible ...

SOLUTIONS
FOR CERTAIN RECTANGULAR SLABS
Equation (73) may in turn be broken up into three parts,
where the parts are chosen as follows:
w2 = w' + w" + w'", (74)
1 - Pa2 1_
w' = - [1 + a (y + v)] e-"(+0) sin au sin ax (75)
3 + p 2r 3 N ,2,3- n 3
which, for y > (-v), is the solution for the infinitely long slab
loaded with a concentrated force (1 - u)P/(3 + u) at the point
x = u, y = -v;
and
1 + p 2Pa 2 1
w" = + - ) N -- e-(Y +v +) sin au sin ax, (76)
(3 + 4) (1 - PN I us... n'
1 - p Pvy 1
w'" = -- e-a(y+v) sin au sin ax. (77)
3 + p rN ,...3 n
For the first part of the correction, w', all of the previous results
for a concentrated load on an infinitely long slab are available. For
the second part, w", it may be observed that
Since
2gw'
0 2 w
v 2 w" = 0.
N-- - N
9x • 9y 2 1 ± + 2P 1
- - - e-"(Y+v)sin au sin ax,
(3 + A) (1 - 1) 7r .... n
one has a relationship similar to (10) and (11), from which
a 2 w" a 2 w" 1 + A P B 1
N- - -N- = loge (78)
Ox 2 Oy 2 (3 + /) (1 - A) 2r A,