Solutions for certain rectangular slabs continuous over flexible ...

ILLINOIS ENGINEERING EXPERIMENT STATION
20. Concentrated Loads on the Slab Giving a Slab Deflection Which
Is Anti-symmetrical With Respect to the Longitudinal Center Line.-
The slab is shown in Fig. 22. The part of the slab in the region of
positive y behaves as if it were a rectangular slab of width b and
length a, simply supported on the three edges x = 0, x = a, y = 0,
supported by an edge beam at y = b, and loaded by a single cona'7
FIG. 22
centrated load P at the point x = u, y = v. For the slab as shown
in Fig. 22 the presence of a supporting beam along the line y = 0
has no effect upon the solution since there is no deflection along
that line.
One may divide the slab into regions and express the deflection
of the slab in these regions by the equations
w1 = w' + w' for v > y > -v (117)
and
w2 = w' + w' for b > y > v. (118)
In these equations w' and wu' are deflections of the infinitely long slab
of span a due to the pair of anti-symmetrical loads shown in Fig. 22,
and w' is the deflection due to corrections applied at the edges y = ± b.
Thus:
, Pa 2 1
0 -- [( 1 + a v ) sinh ay -ay cosh ay] e - " v sin au sin ax,(119)
7"/¥ 1,2,3,.. n,
Pal 1
w = --- - [(1+ay) sinh av - av cosh av] e-
iv ,2..... n
Y
sin au sin ax, (120)
Pa 2 1
w -N
2
-1 (a sinh ay+di cay cosh acy) sin au sin ax. (121)