264
XV
STRESSES IN A RIGHT CIRCULAR CYLINDER
+
F
-=
π
F
+ D
(cos [2(r,p) — (ƒ,p)]d(p,r) += cos (f,m) (d(r,p)
p.) + cos (1. m) (d(r,p)
= F cos (f,m),
which was to be proved.
π
In calculating the first of the parts into which we
separated the integral we ought strictly to have excluded from
the integration the portion of the curved surface lying close
to the element considered; but a simple investigation shows
that the error thus committed is infinitesimal.
Example.-Particular applications of our formula may be
made to cases where pressures are applied at isolated points of
the curved surface. Imagine, for instance, a cylinder placed
between two parallel plane plates which are pressed together
with a pressure P. This is approximately the position of the
rollers which frequently form the basis of support of iron
bridges. We take the axis of x to be the line joining the
points of contact of the cylinder with the planes; its inter-
section with the axis of the cylinder we take as origin. The
coordinate perpendicular to x we call y, and denote by r₁, 2
the distances of the element considered from the points of
contact. Then the component of stress normal to the element
considered is
N₂
2P( cos (r₁x) cos² (r¸n) + cos (r¸x) cos³ (r¸n)
= -
P
Rπ
+
π
Τ1
+
2
T2
If we determine the direction n so that N, becomes a
maximum or minimum, keeping r,, r, the same, we get the
values and directions of the principal stresses at the point
(₁₂). This calculation can be performed. For the axes of
x and y the principal stresses are parallel to the axes, whence
we easily obtain, for the axis of x
X,
=
P 3R2+x²
Rπ R²x²
,
Yy
=
P
RT
?