Solutions for certain rectangular slabs continuous over flexible ...

SOLUTIONS FOR CERTAIN RECTANGULAR
SLABS
Certain other properties of sgn y are important, namely,
and
sgn 2 y - 1
sgn (-y) = -sgn y.
In the analysis of slabs the relationship "y sgn y" frequently
occurs. While this can always be represented by the symbol lyl,
there is no provision in the latter representation for taking the derivative.
It is sufficient, however, to write
\y\ = y sgn y
from which one obtains
d d
y-- \y - (y sgn y) = sgn y.
dy dy
A movement of the origin which results from the substitution of
(y - v) for y, where v is a constant, has the effect, of course, of shifting
the sign change from the line y = 0 to the line y = v.
In a number of the problems in the text use is made of the relationships
just given. For example, in Equation (9) there appears a
function of y, namely,
which may be written in the form
f(y) = (1 + aIy-v|) e-" i -
f(y) = [1 + a (y - v) sgn (y - v)] e-" ( -
)
-* (y-'.
Derivatives of this function with respect to y, when a and v are constants,
are
df
= -a 2 (y - v) [sgn 2 (y - v)] e-"a(y - ) sgn (y-v)
dy
2 "
= -a (y - v) e- -l,
v
d(f= -a [1 - a (y - v) sgn (y - v)] e-
dy 2 = -a 2 (1 - ay-v) e-
1 -
a ,
Y -)
sgn (Y-v)