.
Choosing axes (x1, y1) at the tip of the equilateral triangular cross section
by means show that
M
=
Gβa4
15
√
3
.
C. Weber proposed the following elementary method of examining the effects of a circular
groove or slot in a circular bar [for other kinds of groove and bar combinations, see
Weber and G¨unther (1958)]: Considering a pair of harmonic functions
x
and
x/(x2 +
y2)
obtained from
zn
with
n
= ±1, Weber transformed the functions into polar coordinates
(r, θ). Thus,
x
=
r
cos
θ
and
x/(x2 +
y2)
=
(cos
θ)/r. Hence, he took
where
β
is taken to denote the angle of twist per unit length. Setting
φ
= 0 on the
boundary, Weber obtained the equation of the boundary of the cross section as
(r2 −
b2)
Equation (b) is satisfied identically by the conditions
r2 −
b2 = 0
r
− 2a
cos
θ
= 0
(c)
Equations (c) may be considered to represent the cross section
R
of a circular shaft with
a circular groove . Hence, with Eq. (a), the stress components
τxz, τyz
may
be computed by Derive the formulas for
τxz, τyz.