LIMITS OF STRESS.
327
Substituting in (51), we have
F Fæn
^=J+ —
For any rib of given dimensions, b, p, and I are fixed ; in other words,
they are constants, and the only variables are F and æ.
Re-arranging, we get
F^+æ^^.............................(55)
\pb / p
which is an equation of the form
F(Ct + æ) = C2,
where Cj and C2 are constants. Now, this is the equation of a rectangular
or equilateral hyperbola, one of whose asymptotes is the line L M, and the
other is a line parallel to the longitudinal axis and at a distance,
from it on the inner side. Consequently, if we fix upon a limiting value
for 7, we may vary F and æ within the range shown in fig. 262, where
i + æ constitutes the abscissa, and F the corresponding ordinate for any
p b
point of the hyperbolic curve, Y Z. The point 0 is the intersection of the
asymptotes. Given the distance, æ, from the longitudinal axis of the
resultant, its maximum value is determined by the ordinate ; and vice verså,
given the magnitude of the resultant, the extreme limit of its position
may be deduced.
The diagram, fig. 262, is only applicable to resultants on the compressive
side of the axis. For loci of F between O and M, it would be necessary to
draw another hyperbola, with its origin on the other side of 0; but
instances of this kind do not usually occur in practice and need not
be further considered.