Solutions for certain rectangular slabs continuous over flexible ...

ILLINOIS ENGINEERING EXPERIMENT STATION
9. Concentrated Load at Any Point on the Infinitely Long Slab
Having a Flexible Cross Beam.*-The slab and loading are shown
in Fig. 11. The deflection of the slab without the cross beam is
FiG. 11
given by w 0 of Equation (9). The deflection due to
unknown distribution on the line y = 0 is given by
is to be determined from the boundary condition:
a line load of
(19) where a.
q= E,I = V- - [ = -2N -(vi) .
dX 4 L J =- L ay J =
In this equation q is the downward load on the beam, E 1 and 1I are
the modulus of elasticity and moment of inertia of the beam, respectively,
z is the deflection of the beam given by the equation
z = w] = [WO + Wi ,
V( 1) is the shear due to w 1 , and E is an infinitesimally small positive
distance.
From the application of the boundary condition to (19), one finds
(1 + av) e-
an---- 4 4
(43)
where
nrHi
*Bay obtained solutions for a similar slab supporting either a uniformly distributed load over the
entire slab or a uniform line load directly over the beam. See Hermann Bay, "Die Kreuzweise gespannten
Fahrbahnplatten bei stahlernen Bricken," Beton und Eisen, 33, Heft 14, July, 1934, p. 222-224.
aN