164 DOCK ENGINEERING.
however, to give a succinct description of the method by which the general
formula is evolved.
In fig. 87, let the angle NOM ( = p) represent the limiting angle of
repose, and the semicircle N2 N N0, the locus of the point N, as in fig. 85.
Through O draw the line OX Y, making the angle M OY = a, the
obliquity of the conjugate pressures, and cutting the semicircle in X and Y.
Then the limits of the ratio of the intensities of the conjugate pressures are
OX ,0Y
OY and OX’
The angle a may have any value between zéro and p. In the former
limit, which is the case when the conjugate pressures are perpendicular to
each other, and become principal stresses, 0 X Y coineides with O N2 N0 and
9 -A- ( - I -sni p\ .s the mjnimum value of -. When the obliquity is
0 N0 ( 1 + sin p) q
the greatest possible, such that a = p, the points N2 and N0 coalesce in
N, and the limit of the ratio of the conjugale pressures becomes unity.
For any intermediate position in which a = X O M, the limiting ratio
(^ of the conjugale pressures may be determined as follows :—Draw
S M perpendicular to X Y, and join M X, M Y, each line making the
angle ô with X Y.
p' OX_OS-XS_| (q+p) cos a-^(q-p) cos ô
en q' OY OS+YS J (q+p) cos a+J (q-p) cos 6
q+P
cos a - cos 0
= q-p
q+p A
-— - cos a + cos Ô
q-P
• (14>
Now,
• A 1 (q+p) •
sin ^= — — sin a,
i (q-p
.•. cos Ô=
sin- a
And as
sin p=
^(q-p)
i (q+p)’
COS 0=
l2 sin2 p - (q+p)2 sin2 a
(q - P)2
—L^!L ^gjn2 0_ gin2 œ
q-p
=-^ Jcos- a - cos2 p.
q-p