Solutions for certain rectangular slabs continuous over flexible ...

18 ILLINOIS ENGINEERING EXPERIMENT STATION
The equations for moment then become
1 + p 1 - • 9(0 "
M9 = +- s + (y - v)
2 2 ay
1 + 1 - ( ) 900
MY =- -
0 (y v) -
(16)
2 2 ay
M> =
1 -
(y - v) 0€o
2 Ox
Substitution of 40 and its derivatives into (16) gives the following
equations for the moments at any point x, y:
M° (1+•)P Bo (l-s )P(y-v) 1 1 ir(y--v)
M - - l o g e Ao- --- 8+- -) s--
MY 8r Ao 8a BO A) a
(1-_ )P(y-v) 1r . (x-u) 1 . r(x+u)-1
MY = - -- smin '-sin .
8a Bo a A a J
l(17)
Near the load the moments given by these equations are not valid.
The special formulas given by Westergaard* may be used to obtain
bending moments which lead to the proper maximum stresses under
the load. With the origin of co6rdinates and the position of the load
given by Fig. 2, the special formulas for the resultant moments under
the load become
S ( 4- ) P / 4a - \ P
M = log - sin-- + -
Sxu 47r ri a / 4
yV-
M(0) M?]1 -(I P
x=u J ,u 47
- (18)
XM( = 0,
*H. M. Westergaard, Computation of Stresses in Bridge Slabs Due to Wheel Loads, Public Roads,
V. 11, No. 1, March, 1930, p. 8. See Westergaard's Equations 57 to 62 inclusive.
Holl gives special treatment to the problem of finding the correct stresses in the vicinity of a
concentrated load on a rectangular slab with pinned-free edges. Use is made of the analysis of a thick
square slab with simply supported edges. See D. L. Holl, Analysis of Thin Rectangular Plates Supported
on Opposite Edges, Bulletin 129, Iowa Engineering Experiment Station, Iowa State College,
1936, p. 34.