182
6 Two-Dimensional Problems in Elasticity …
(σ r ) max =
3 + ν
8
ρw
2
(b − a)
2
(6.52)
The maximum circumferential stress is at the inner boundary, where
(σ θ ) max =
3 + ν
4
ρw
2
b
2
+
1 − ν
3 + ν
a
2
(6.53)
The displacement u r for all the cases considered can be calculated as below:
u r = r ε θ =
r
E
(σ θ − νσ r )
(6.54)
6.10 Stress Concentration
While discussing the case of simple tension and compression, it has been assumed
that the bar has a prismatical form. Then for centrally applied forces, the stress at
some distance from the ends is uniformly distributed over the cross-section. Abrupt
changes in cross-section give rise to great irregularities in stress distribution. These
irregularities are of particular importance in the design of machine parts subjected
to variable external forces and to reversal of stresses. If there exists in the structural
or machine element a discontinuity that interrupts the stress path, the stress at that
discontinuity may be considerably greater than the nominal stress on the section; thus
there is a “stress concentration” at the discontinuity. The ratio of the maximum stress
to the nominal stress on the section is known as the ‘Stress Concentration Factor’.
Thus, the expression for the maximum normal stress in a centrically loaded member
becomes
σ = K
P
A
(6.55)
where A is either gross or net area (area at the reduced section), K = stress concentration factor and P is the applied load on the member. In Fig. 6.8a, b, c, the type of
discontinuity is shown and in Fig. 6.8d, e, f, the approximate distribution of normal
stress on a transverse plane is shown.
Stress concentration is a matter, which is frequently overlooked by designers. The
high stress concentration found at the edge of a hole is of great practical importance.
As an example, holes in ships decks may be mentioned. When the hull of a ship
is bent, tension or compression is produced in the decks and there is a high stress
concentration at the holes. Under the cycles of stress produced by waves, fatigue of
the metal at the overstressed portions may result finally in fatigue cracks.
6 Two-Dimensional Problems in Elasticity …
(σ r ) max =
3 + ν
8
ρw
2
(b − a)
2
(6.52)
The maximum circumferential stress is at the inner boundary, where
(σ θ ) max =
3 + ν
4
ρw
2
b
2
+
1 − ν
3 + ν
a
2
(6.53)
The displacement u r for all the cases considered can be calculated as below:
u r = r ε θ =
r
E
(σ θ − νσ r )
(6.54)
6.10 Stress Concentration
While discussing the case of simple tension and compression, it has been assumed
that the bar has a prismatical form. Then for centrally applied forces, the stress at
some distance from the ends is uniformly distributed over the cross-section. Abrupt
changes in cross-section give rise to great irregularities in stress distribution. These
irregularities are of particular importance in the design of machine parts subjected
to variable external forces and to reversal of stresses. If there exists in the structural
or machine element a discontinuity that interrupts the stress path, the stress at that
discontinuity may be considerably greater than the nominal stress on the section; thus
there is a “stress concentration” at the discontinuity. The ratio of the maximum stress
to the nominal stress on the section is known as the ‘Stress Concentration Factor’.
Thus, the expression for the maximum normal stress in a centrically loaded member
becomes
σ = K
P
A
(6.55)
where A is either gross or net area (area at the reduced section), K = stress concentration factor and P is the applied load on the member. In Fig. 6.8a, b, c, the type of
discontinuity is shown and in Fig. 6.8d, e, f, the approximate distribution of normal
stress on a transverse plane is shown.
Stress concentration is a matter, which is frequently overlooked by designers. The
high stress concentration found at the edge of a hole is of great practical importance.
As an example, holes in ships decks may be mentioned. When the hull of a ship
is bent, tension or compression is produced in the decks and there is a high stress
concentration at the holes. Under the cycles of stress produced by waves, fatigue of
the metal at the overstressed portions may result finally in fatigue cracks.