Solutions for certain rectangular slabs continuous over flexible ...

SOLUTIONS FOR CERTAIN RECTANGULAR
SLABS
and
w2 = w"' + w' for b > y > v (93)
where wo and wo' represent the deflections, in their respective regions,
of the infinitely long slab of span a due to the pair of concentrated
loads shown in the figure, and where w' represents the deflection due
to corrective moments and shears which are applied to the slab along
the edges y = +b in order to produce the proper edge conditions.
From the solutions given previously, one finds, where v>y> -v,
Pa 2 1
w'= N [(1 +av) cosh ay - ay sinh ay] e - v sin au sin ax (94)
VN ,s,.. n 3
and, where b > y > v,
Pa 2 1
w'P =-- [(l+ay) cosh av-av sinh av] e-" sin au sin ax. (95)
irWN 7- n'
The deflection due to the symmetrical edge corrections may be written
in the form
Pa 2 1
w' = - . - (b, cosh ay - cl ay sinh ay) sin au sin ax (96)
where bi and ci are parameters to be determined by the boundary
conditions.
The boundary conditions are
_- + _ -w = 0, (97)
yx a 2 yb_
d9z w 2 0 w 2 2
8
q = E2 2 - = N -- + ( 2 -M) , (98)
dx 4
3
y=b
x 2 y -b
where q is the downward load on the beam and z is the downward
deflection. Substitution of the derivatives obtained from (95) and
(96) into (97) and (98) results in two equations in bi and Ci whose
simultaneous solution gives