Solutions for certain rectangular slabs continuous over flexible ...

60 ILLINOIS ENGINEERING EXPERIMENT STATION
over the stiffened edges and the total deflection, valid over the entire
area of slab, becomes
Pa 2 1
w = - (F 2 sinh ay - F 4 ay cosh ay) sin au sin ax (127)
where F 2 and F 4 are quantities given in Appendix B.
The deflection of the beam at y = b becomes
P a 2 1
z2 = w = , - F 7 sin au sin ax
=bi T7r3N 1.2,- n 3 (128)
2Pa 3 1
= zo -- - E -- F 8 sin au sin ax,
74E2I 1,2,3,.. n 4
where zo is the simple beam deflection given by (115) and where F 7
and F 8 are functions stated in Appendix B.
The bending moment in the beam at y = b is
d'zo 2Pa 1
Mbeam = -E - - , , -- Fs sin au sin ax. (129)
dax 2 72 12,3,. n 2
This moment is a maximum under the load when u = a/2. Then
Pa 8 Fs
max. Mbeam = -- 1 -- . (130)
4 7r2 i ,,.. n
As before, the bending moment M, at the edge of the slab may be
found from the moment in the beam by the relation
S 1 - p2
Mx b - Mbeam.
22. Load Uniformly Distributed Over the Entire Slab.*-When the
intensity of load is p per unit of area and the axes are chosen as shown
*This problem has been treated by others. See, for example, Emil Mtiller, "Uber rechteckige
Platten, die lings zweier gegenuiberliegenden Seiten auf biegsamen Tragern ruhen," Zeit. fur angew.
Math. und Mech., 6, 1926, p. 355-66. Miuller gives numerical values of bending moments at the center
of the slab, assuming that Poisson's ratio I = 0, for ratios of b/a varying from 1/8 to 1.0, and for
ratios of EJ 1 /(bN) of 0, 0.5, 1.0, 1.5, 2.0, 2.5 and 3.0. See also B. G. Galerkin, "Elastic Thin Plates,"
Gosstrojisdat, Leningrad and Moscow, 1933, (In Russian) p. 86.