Solutions for certain rectangular slabs continuous over flexible ...

SOLUTIONS FOR CERTAIN RECTANGULAR
SLABS
The solution represented by (217) is satisfactory to use when the
ratio b/a is not too small. When the ratio b/a is small, in particular
when it is so small that the effect of the supports at y = ± a/2
becomes negligible, a bending moment obtained from (217) is in
the form of a series whose convergence is very slow, and whose
terms may be made up of small differences of large numbers. This
difficulty may be avoided by putting wi in a special form in the
following manner:
The deflection, wi, may be written as the sum of two parts, wo
and wor, where w 0 is the deflection of the rectangular slab, b by a,
simply supported on four edges, and loaded by a concentrated force P
at the center. The remaining part, wcor, is a correction to be added
to w 0 to give the effect of the continuity of the slab across the beams
at x = +b/2. From Equation (109), with the modifications resulting
from the position of the load and from the orientation of the
axes, one finds
Pa 2 1
wo = - - (f cosh ax - sinh ax
27rN ..... " n3 (219)
+ ax cosh ax - f/ ax sinh ax) cos ay
where
sinh ab - ab sinh ab
f l =
,T f = -
( 220)
1 + cosh ab 1 + cosh ab
The corrective deflection function is found as the difference
between wi and w 0 . Thus the corrective deflection function is
Wcor W- 1 WO
Pab 1 ab ab (2 2 1 )
=- - k - tanh - cosh ax - ax sinh ax cos ay
2Tr 2 N ni,... \2 2
where
ab
tanh - 2
k 5 = 1 + b + sinh b + cosh b (222)
1 + ab + sinh ab + cosh ab