Solutions for certain rectangular slabs continuous over flexible ...

SOLUTIONS FOR CERTAIN RECTANGULAR
SLABS
Re = concentrated reaction at corner, as shown in Fig. 1
or,, o- = normal components of stress, in directions of x and y
respectively
Txv = shearing component of stress, in plane normal to x and
in direction of y, or in plane normal to y and in direction
of x
n = an integer. A particular summation is to be carried out
for the integral values indicated under the summation
sign
n7r
a =
a
nirb
3 = ab = -
a
= an infinitesimally small positive quantity or distance
)2 a 2
V 2 - + - , Laplace's operator in two variables
X2 a9-y 2
Various other quantities are defined in the text where they are
used or in Appendix B.
II. REVIEW OF FUNDAMENTAL EQUATIONS
4. The Ordinary Theory of Slabs.-The equations pertaining to the
ordinary theory of flexure of slabs will not be derived here since their
derivation is available in a number of places* in the technical literature.
It should be stated, however, that the ordinary theory is based
upon the assumption that every line drawn through the slab normal
to its middle surface before the slab is loaded remains straight and
normal to the deflected middle surface after the slab is loaded. The
theoretical results obtained under this assumption are applicable to
bridge slabs except in the vicinity of a concentrated load.
NAdait found the maximum stress under a loaded circular area
by a special theory applicable to a thick circular slab and Westergaardt
converted NAdai's results into expressions like those from the
ordinary theory by the use of an "equivalent diameter" of a uniformly
loaded circular area. These results are valid when the load is sufficiently
remote from a support, as it must be to produce maximum
*See, for example: A. Nadai, Die elastischen Platten, 1925, p. 20; H. M. Westergaard and W. A.
Slater, Moments and Stresses in Slabs, Proc. Am. Cone. Inst., V. 17, 1921, p. 415; H. M. Westergaard,
Computation of Stresses in Bridge Slabs Due to Wheel Loads, Public Roads, V. 11, No. 1, March,
1930, p. 2.
tA. Nadai, Die elastischen Platten, 1925, p. 308.
$H. M. Westergaard, Stresses in Concrete Pavements Computed by Theoretical Analysis, Public
Roads, V. 7, No. 2, April, 1926, p. 25.