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Chapter 14: Stress concentrations
Locally high stresses can arise due to
Abrupt changes in section properties (hole, corner)
Contact stresses (bearing, gear, etc)
Material discontinuities
Initial stresses due to manufacturing process
Cracks
Structure is often designed without considering
them followed by local fixes Instead we often assume some
defects of given
size as safety measure
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Stress concentration factor
n
cS
max=
Maximum over average
(nominal stress)
Nominal stress
n
P
=
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Analytical solution for circular hole
Circular hole in infinite plate
under uniaxial load (Airy stress
function in polar coordinates)
For r=a
What are the other stresses atthe boundary?
2 2 2
2 2 2
2 4
2 4
2 2
2 2
31 1 1 cos 2
2 2
31 1 cos 2
2 2
31 1 cos 2
2
rr
r
a a a
r r r
a a
r r
a a
r r
= +
= + +
= +
)2cos21( =
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For circular hole can find analytical
solution At = /2
Maximum at r= a
What will happen if stresses are applied both x-
and y-directions? What would be the worst combination of
stresses?
2 4
2 42 3
2a ar r
= + +
(max) ( ) 3r a = = =
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Elliptic hole
Where curvilinear coordinates pay off
Infinite plate stressed perpendicular to major axis
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Elliptical coordinates Elliptic coordinates (, )
Equation for ellipse
What is similar to polar coordinates? FE models?
2
2
2
2
2
sinhcoshc
yx=+
00 sinhcosh cbca ==
0 -> 0: elliptic holebecomes a sharp crack
a -> c (crack length)
b-> 0
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Stress concentration for ellipse
Stress components:
At the hole surface (= 0, = 0):
Max stress at = 0:
+
=+
12cos2cosh
2sinh)1( 00
22
ee
+=
= 12cos2cosh
2sinh)1(
0
0
22
0
0
0
ee
0tanh
b
a
=
(max) 0
2(1 2 coth ) 1
a
b
= + = +
What will happen for
a circular hole?
Crack?
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SCF for grooves, holes, and fillets
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Grooves and holes
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Reading assignment
Sections 15.1-2: For a thin-wall cylinder what combinations
of
loading and cracks would produce mode I, mode II, and
mode III fracture?
Source: www.library.veryhelpful.co.uk/ Page11.htm
