1.20 Saint-Venant Principle
21
1.19 Phenomenon of Stress Concentration
Assume that in a cylinder considered in the previous paragraph p a = 0, b → ∞.
In this case, by dividing numerators and denominators of formula (1.35) by b 2 and
then making a passage to the limit with b → ∞, we will find
σ t (r) = −p b − p b
a 2
r 2 , (p a = 0, b = ∞).
(1.37)
The first term in the right part of the last formula represents annular stress that
would be present in the case of all-around equal compression of a continuous disc.
The second term expresses the change in annular stresses that is caused by the
appearance of a round hole with a radius of a, which is small as compared to the
disc dimensions. We have σ t (a) = −2p b on the outline of the hole, e. g. the stress
on the hole outline is twice as much as in those that could have appeared in the
continuous disc. The same result for σ t (a) will be obtained for final b if we bring
the hole radius (a) to zero, e. g.
lim
a→0
σ t (a) = −2p b , (p a = 0),
(1.38)
whereas for a = 0, formula (1.35) gives σ t = −p b .
This paradox is explained by the fact that there is no specific bond in sufficiently
small volumes between internal forces in a real body and in an ideally homogeneous
model. For real internal forces, the influence of the hole at a → 0 disappears despite
formula (1.38), since there is no difference in the micro-structure of a continuous
disc and the same disc with a sufficiently small hole.
A rise in stresses near relatively small holes and cut-outs on the body surface is
called the stress concentration.
1.20 Saint-Venant Principle
Assume p b = 0, b → ∞ in formulas (1.34) and (1.35). We obtain
σ r = −
a 2
r 2 p a , σ t =
a 2
r 2 p a .
Stress from internal pressure in the case of r a becomes relatively low. By
summarizing this observation in any case of a plane stress-strain state, we come to
the following conclusion:
if a system of mutually equilibrating forces is applied in a specific area, stresses from such
loading have a substantial value only in such vicinity of the loaded part of the body whose
dimensions are commensurable with the dimensions of the loaded area.
21
1.19 Phenomenon of Stress Concentration
Assume that in a cylinder considered in the previous paragraph p a = 0, b → ∞.
In this case, by dividing numerators and denominators of formula (1.35) by b 2 and
then making a passage to the limit with b → ∞, we will find
σ t (r) = −p b − p b
a 2
r 2 , (p a = 0, b = ∞).
(1.37)
The first term in the right part of the last formula represents annular stress that
would be present in the case of all-around equal compression of a continuous disc.
The second term expresses the change in annular stresses that is caused by the
appearance of a round hole with a radius of a, which is small as compared to the
disc dimensions. We have σ t (a) = −2p b on the outline of the hole, e. g. the stress
on the hole outline is twice as much as in those that could have appeared in the
continuous disc. The same result for σ t (a) will be obtained for final b if we bring
the hole radius (a) to zero, e. g.
lim
a→0
σ t (a) = −2p b , (p a = 0),
(1.38)
whereas for a = 0, formula (1.35) gives σ t = −p b .
This paradox is explained by the fact that there is no specific bond in sufficiently
small volumes between internal forces in a real body and in an ideally homogeneous
model. For real internal forces, the influence of the hole at a → 0 disappears despite
formula (1.38), since there is no difference in the micro-structure of a continuous
disc and the same disc with a sufficiently small hole.
A rise in stresses near relatively small holes and cut-outs on the body surface is
called the stress concentration.
1.20 Saint-Venant Principle
Assume p b = 0, b → ∞ in formulas (1.34) and (1.35). We obtain
σ r = −
a 2
r 2 p a , σ t =
a 2
r 2 p a .
Stress from internal pressure in the case of r a becomes relatively low. By
summarizing this observation in any case of a plane stress-strain state, we come to
the following conclusion:
if a system of mutually equilibrating forces is applied in a specific area, stresses from such
loading have a substantial value only in such vicinity of the loaded part of the body whose
dimensions are commensurable with the dimensions of the loaded area.