6.13 Winkler–Bach Theory
195
3. With the above sign convention, if σ θ is positive, it denotes tensile stress while
negative sign means compressive stress.
The distance between the centroidal axis (y = 0) and the neutral axis is found by
setting the tangential stress to zero in Eq. (6.64)
∴ 0 =
M
AR
1 +
y
m(R + y)
or 1 = −
y n
m(R−y n )
.
where y n denotes the distance between axes as indicated in Fig. 6.11. From the
above,
y n = −
m R
(m + 1)
This expression is valid for pure bending only.
However, when the beam is acted upon by a normal load P acting through the
centroid of cross-sectional area A, the tangential stress given by Eq. (6.64) is added
to the stress produced by this normal load P. Therefore, for this simple case of
superposition, we have
σ θ =
P
A
+
M
AR
1 +
y
m(R + y)
(6.65)
As before, a negative sign is associated with a compressive load P.
6.14 Stresses in Closed Rings
Crane hook, split rings are the curved beams that are unstrained at one end or both
ends. For such beams, the bending moment at any section can be calculated by
applying the equations of statics directly. But for the beams having restrained or fixed
ends such as a close ring, equations of equilibrium are not sufficient to obtain the
solution, as these beams are statically indeterminate. In such beams, elastic behaviour
of the beam is considered and an additional condition by considering the deformation
of the member under given load is developed as in the case of statically indeterminate
straight beam.
Now, consider a closed ring shown in Fig. 6.12a, which is subjected to a
concentrated load P along a vertical diametrical plane.
The distribution of stress in upper half of the ring will be same as that in the lower
half due to the symmetry of the ring. Also, the stress distribution in any one quadrant
will be same as in another. However, for the purposes of analysis, let us consider a
quadrant of the circular ring as shown in Fig. 6.12c, which may be considered to be
195
3. With the above sign convention, if σ θ is positive, it denotes tensile stress while
negative sign means compressive stress.
The distance between the centroidal axis (y = 0) and the neutral axis is found by
setting the tangential stress to zero in Eq. (6.64)
∴ 0 =
M
AR
1 +
y
m(R + y)
or 1 = −
y n
m(R−y n )
.
where y n denotes the distance between axes as indicated in Fig. 6.11. From the
above,
y n = −
m R
(m + 1)
This expression is valid for pure bending only.
However, when the beam is acted upon by a normal load P acting through the
centroid of cross-sectional area A, the tangential stress given by Eq. (6.64) is added
to the stress produced by this normal load P. Therefore, for this simple case of
superposition, we have
σ θ =
P
A
+
M
AR
1 +
y
m(R + y)
(6.65)
As before, a negative sign is associated with a compressive load P.
6.14 Stresses in Closed Rings
Crane hook, split rings are the curved beams that are unstrained at one end or both
ends. For such beams, the bending moment at any section can be calculated by
applying the equations of statics directly. But for the beams having restrained or fixed
ends such as a close ring, equations of equilibrium are not sufficient to obtain the
solution, as these beams are statically indeterminate. In such beams, elastic behaviour
of the beam is considered and an additional condition by considering the deformation
of the member under given load is developed as in the case of statically indeterminate
straight beam.
Now, consider a closed ring shown in Fig. 6.12a, which is subjected to a
concentrated load P along a vertical diametrical plane.
The distribution of stress in upper half of the ring will be same as that in the lower
half due to the symmetry of the ring. Also, the stress distribution in any one quadrant
will be same as in another. However, for the purposes of analysis, let us consider a
quadrant of the circular ring as shown in Fig. 6.12c, which may be considered to be