Lecture Slides
Chapter 5
Failures Resulting from
Static Loading
© 2015 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied,
scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter Outline
5-1 Static Strength
If experimental test data is not available, use published
strength values.
Experimental test data is generally warranted for large
quantities or when failure is costly (in time, expense, or
life).
Methods are needed to safely and efficiently use
published strength values for a variety of situations.
}
}
}
3
University of New Orleans
5-2 Stress Concentration
Graphs are available
for standard
configurations (A-15,
16).
Many more are
available
Note the trend for
higher Kt at sharper
discontinuity radius,
and at greater
disruption
}
}
}
4
University of New Orleans
5-2 Stress Concentration
}
For ductile materials:
} Use Kt as given in the appendix for dynamic loading
} Use Kt = 1 for static loading
}
For brittle materials:
} Use Kt as given in the appendix for dynamic and static
loading
5
University of New Orleans
5-3 Failure Theories
Need for failure theories:
} One strength, multiple stresses
} How to compare stress states to a single strength?
}
}
6
Failure theories propose a means to compare multiaxis stresses to a single value of strength.
University of New Orleans
5-3 Failure Theories
}
Five failure theories are commonly used. Recall that a
ductile material has εf > 0.05, and a brittle material has εf
< 0.05.
}
Ductile Materials
•
Maximum shear stress (MSS)
•
Distortion energy (DE) [von Mises-Hencky, shear-energy, octahedral
shear stress]
•
Ductile Coulomb-Mohr (DCM)
Brittle Materials
}
7
•
Maximum normal stress (MNS)
•
Brittle Coulomb-Mohr (BCM)
University of New Orleans
5-3 Why Different Theories?
8
University of New Orleans
5-3 Why Different Theories?
9
University of New Orleans
5-3 Failure Theories
}
Which stress causes failure?
} Ductile materials are limited by their shear strength
} Brittle materials are limited by their tensile strength
}
How to define failure?
} Fracture is an obvious answer, but yielding can be
considered a failure if it distorts the machine element
so much that its function cannot be performed.
10
University of New Orleans
5-4 Maximum Shear Stress
}
Five failure theories are commonly used. Recall that a
ductile material has εf > 0.05, and a brittle material has εf
< 0.05.
11
University of New Orleans
5-4 Maximum Shear Stress
12
University of New Orleans
5-4 Maximum Shear Stress
For plane stress states, three
cases define the failure
surface:
}
Case 1: σA ≥ σB ≥ 0
} σA ≥ Sy
}
Case 2: σA ≥ 0 ≥ σB
} σA − σB ≥ Sy
}
Case 3: 0 ≥ σA ≥ σB
} σB ≤ −Sy
13
University of New Orleans
5-5 Distortion Energy Theory
}
}
Experiments show ductile materials stressed
hydrostatically exhibited strengths greatly in excess of
expected values.
The DE theory claims if strain energy is divided into
volume changing energy and distortion energy, the failure
is primarily affected by the distortion energy.
14
University of New Orleans
5-5 Distortion Energy Theory
}
Theory: Yielding occurs when the distortion strain energy
reaches the yield strength in simple tension or
compression.
15
University of New Orleans
5-6 Distortion Energy Theory
}
The strain energy per unit volume for a volume change
only is
2
3σ av
(1− 2ν ) 1− 2ν $ 2
uv =
=
σ 1 + σ 22 + σ 32 + 2(σ 1σ 2 + σ 2σ 3 + σ 3σ 1) %
'
2E
6E &
}
The strain energy per unit volume for distortion only is
2
2$
2
#
σ
−
σ
+
σ
−
σ
+
σ
−
σ
1+ν ( 1 2 ) ( 2 3 ) ( 3 1 )
ud = u − uv =
&
'
2
3E &
')
(
16
University of New Orleans
5-6 Distortion Energy Theory
}
For a simple tension test σ1 = Sy & σ2 = σ3 = 0, or
ud =
}
1+ν 2
Sy
3E
Yielding is predicted to occur if
1/ 2
" (σ − σ )2 + (σ − σ )2 + (σ − σ )2 #
2
3
3
1
σ'= & 1 2
'
2
&(
')
}
≥ Sy
(5-10)
The term σ’ is an equivalent or effective stress for the
entire state of stress. This effective stress is often called
the von Mises stress.
17
University of New Orleans
5-6 Distortion Energy Theory
}
}
Von Mises stress can be considered a single, equivalent,
or effective stress for the entire stress state.
Distortion Energy failure theory simply compares the Von
Mises stress to the yield strength,
σ ' = Sy
}
Introducing a design factor, the DE theory becomes,
σ'=
18
Sy
n
University of New Orleans
5-6 Distortion Energy Theory
}
For a plane stress state (2 nonzero principal stresses),
(
σ ' = (σ
σ'=
}
σ A2 − σ Aσ B
2
x
2 1/ 2
+σ B
− σ xσ y + σ y2
(5-13)
)
+
2 1/ 2
3τ xy
)
(5-15)
For xyz components of 3 dimensional stress,
1 #
σ'=
σ x −σ y
&
2(
(
2
2
2
) + (σ y −σ z ) + (σ z −σ x )
(
+6
2
τ xy
2
+ τ yz
2
+ τ zx
1/ 2
$
')
)
(5-14)
19
University of New Orleans
5-6 Distortion Energy Theory
}
The plane stress form
of the DE theory is a
rotated ellipse. The
dotted lines represent
the maximum shear
stress theory, which is
more conservative than
the distortion energy
theory.
20
University of New Orleans
5-6 Distortion Energy Theory
}
The distortion-energy static failure theory is also called:
} The von Mises theory
} The shear-energy theory
} The octahedral-shear-stress theory.
21
University of New Orleans
5-6 Distortion Energy Theory
}
For pure shear
loading, the DE
theory predicts:
2 = S
3τ xy
y
τ xy =
Sy
= 0.577 S y
3
Ssy = 0.577 S y
}
For pure shear
loading, the MSS
theory predicts:
22
Ssy = 0.5S y
University of New Orleans
5-6 DE Failure Theory Example
}
For the following principal
stresses, estimate the factor
of safety if the yield strength
is 100 kpsi:
} a) 70, 70, 0 kpsi
} b) 30, 70, 0 kpsi
} c) 0, 70, -30 kpsi
} d) 0, -30, -70 kpsi
} e) 30, 30, 30 kpsi
For case a)
2
σ ' = "70 − 70(70) + 70
n=
23
%
Sy
σ'
=
2 #1/ 2
&
= 70 kpsi
100
= 1.43
70
University of New Orleans
5-6 DE Failure Theory Example
For case b)
2
σ ' = "30 − 30(70) + 70
%
σ ' = 60.8 kpsi
S y 100
n=
=
= 1.64
σ ' 60.8
2 #1/ 2
&
For case c)
2
σ ' = "70 − 70(−30) + (−30)
%
σ ' = 88.9 kpsi
S
100
n= y =
= 1.13
σ ' 88.9
24
2 #1/ 2
&
University of New Orleans
5-6 DE Failure Theory Example
For case d)
1/2
2
2$
"
σ ' = (−70) −(−30)(−70)+(−70)
#
%
σ ' = 60.8kpsi
S y 100
n=
=
= 1.64
σ ' 60.8
For case e)
" (30−30)2 +(30−30)2 +(30−30)2 $1/2
σ'=&
'
2
&#
'%
σ ' = 0.0 kpsi
S y 100
n=
=
=∞
σ'
0
25
University of New Orleans
5-7 DE Failure Theory Example
If the cantilever shaft is made of
AISI 1035 steel (Sy = 81 ksi),
using the D.E. theory, how large
can F be for n = 1?
σ x = Mc = 32Fl3AC = 32*14*3F
I
πd
π (1)
σ x = 142.6 F
Tr 16 FlDC 16*15*F
τ xy =
=
=
= 76.4 F
3
3
J
πd
π (1)
σ'=
(
σ x2
2
+ 3τ xy
1
)
2
1/ 2
2
2
= $(142.6 F ) + 3(76.4 F ) %
&(
')
σ ' = 194.5 F
81000
81000
n=
=1 ⇒ F =
= 416 lb
194.5 F
194.5(1)
26
University of New Orleans
5-6 Coulomb-Mohr Theory
}
}
Some materials have
different tensile and
compressive
strengths.
The Coulomb-Mohr
theory is applicable
to such materials.
27
University of New Orleans
5-6 Coulomb-Mohr Theory
}
The Coulomb-Mohr theory is given as:
}
For ductile materials, use tensile and compressive yield
strengths
For brittle materials, use tensile and compressive ultimate
strengths
}
28
University of New Orleans
Industry Standards and Codes
29
University of New Orleans
5-7 Ductile Materials Summary
}
Shown is a
comparison
between actual
failure and
predicted failure
}
It is clear that the
maximum normal
stress theory is the
least accurate
(especially in the 4th
quadrant) for ductile
materials but predicts
the failure of brittle
materials well.
30
University of New Orleans
5-8 Maximum Normal Stress Theory
}
}
Use for brittle materials the maximum normal
stress theory predicts
that failure occurs
whenever one of the
principal stresses equals
or exceeds the strength.
This theory states that
failure occurs when,
σ1 ≥ Sut
}
or
σ 3 ≤ − Suc
Incorporating a design
factor,
S
−S
σ 1 = ut
or σ 3 = uc
n
n
31
University of New Orleans
5-9 Mohr Theory for Brittle Materials
}
}
Extend the previously
used Coulomb-Mohr
theory for brittle
materials (Eq. 5-31).
The modified Mohr
theory (Eq. 5-32) was
developed to better fit
the data in the 4th
quadrant.
32
University of New Orleans
5-10 Brittle Material Summary
}
Shown is a comparison
between actual and
predicted failure data for
cast iron using four brittle
material failure theories.
}
Notice the modified Mohr
theory tracks the data in
the 4th quadrant very well.
}
Points A, B, C, and D
show that none of the
theories work well in the
3rd quadrant. Testing
should be done if the
loading is in the 3rd
quadrant.
33
University of New Orleans