Solutions for certain rectangular slabs continuous over flexible ...

ILLINOIS ENGINEERING EXPERIMENT STATION
Equation (221), which gives the effect of continuity over the supports
at x = ± b/2, may be compared with the equation obtained
by Westergaard* for the effect of fixing the edges at x = ±b/2.
Westergaard's equation for the change of deflection caused by fixing
the two edges at x = + b/2, in the present notation, may be written
in the form
ab
tanh -
Pab 1 2 (ab ab
Wcor = - - - - tanh - cosh ax
27r 2 N ... n 2 ab + sinh ab 2 2 (223)
- ax sinh ax) cos ay.
The two results are found to be identical except for the presence of
the quantity (1 + cosh ab) in the denominator of the parameter ks.
That is, the substitution of
ab
tanh -
2
k ab + sinh ab (224)
for k 5 in (221) changes the effect from that due to continuity with a
slab of large extent in the x direction to that due to fixing the slab
against rotation and deflection on the lines x = + b/2.
The corrective moments under the load, obtained from (221), are
1 Pb F ab ab]
M7 =- E k 5 2 - (1 - )-- tanh--,
X==o 2a [,5,(. 22 (225)
SPb ab ab1
MC - E k, 2 + (1 - u) - tanh - ,
xMy=0o 2a - 2 2 J
in which k 5 is given by (222).
As in Section 38, the edges at y = +a/2 may be moved sufficiently
far from the load that their effect on the moments in the
vicinity of the load practically disappears. For the initial deflection,
*H. M. Westergaard, Computation of Stresses in Bridge Slabs Due to Wheel Loads, Public Roads,
V. 11, No. 1, March, 1930, p. 20; see Equation 101.