Formula Sheet: Theories of Failure | Solid Mechanics - Mechanical Engineering PDF Download

 Page 1


F ormula Sheet: Theories of F ailure
Introduction to Theories of F ailure
• Definition : Theories of failure predict the conditions under which a mate-
rial fails under complex loading, used to design components safely .
• Applications : Primarily for ductile (e.g., steel) and brittle (e.g., cast iron)
materials under static or dynamic loading.
• Principal Stresses : F or a 3D stress state, principal stresses (s
1
,s
2
,s
3
) are
calculated, t ypically ordered as s
1
= s
2
= s
3
.
F ailure Theories for Ductile Materials
• Maximum Shea r Stress Theory (Tresca) :
s
1
-s
3
= s
y
where s
y
= yield strength.
• F actor o f Safety (FOS) :
FOS =
s
y
s
1
-s
3
• von Mises (Distortion Energy) Theory :
v
(s
1
-s
2
)
2
+(s
2
-s
3
)
2
+(s
3
-s
1
)
2
2
= s
y
or
s
von Mis es
=
v
s
2
x
+s
2
y
-s
x
s
y
+3t
2
xy
= s
y
where s
x
,s
y
= normal stresses, t
xy
= shear stress.
• FOS for v on Mises :
FOS =
s
y
s
von Mises
F ailure Theories for Brittle Materials
• Maximum Nor mal Stress (Rankine) Theory :
s
1
= s
ut
or |s
3
|= s
uc
where s
ut
= ultimate tensile strength, s
uc
= ultimate compressive strength.
• FOS :
FOS =
s
ut
s
1
(tension) or FOS =
s
uc
|s
3
|
(compression)
1
Page 2


F ormula Sheet: Theories of F ailure
Introduction to Theories of F ailure
• Definition : Theories of failure predict the conditions under which a mate-
rial fails under complex loading, used to design components safely .
• Applications : Primarily for ductile (e.g., steel) and brittle (e.g., cast iron)
materials under static or dynamic loading.
• Principal Stresses : F or a 3D stress state, principal stresses (s
1
,s
2
,s
3
) are
calculated, t ypically ordered as s
1
= s
2
= s
3
.
F ailure Theories for Ductile Materials
• Maximum Shea r Stress Theory (Tresca) :
s
1
-s
3
= s
y
where s
y
= yield strength.
• F actor o f Safety (FOS) :
FOS =
s
y
s
1
-s
3
• von Mises (Distortion Energy) Theory :
v
(s
1
-s
2
)
2
+(s
2
-s
3
)
2
+(s
3
-s
1
)
2
2
= s
y
or
s
von Mis es
=
v
s
2
x
+s
2
y
-s
x
s
y
+3t
2
xy
= s
y
where s
x
,s
y
= normal stresses, t
xy
= shear stress.
• FOS for v on Mises :
FOS =
s
y
s
von Mises
F ailure Theories for Brittle Materials
• Maximum Nor mal Stress (Rankine) Theory :
s
1
= s
ut
or |s
3
|= s
uc
where s
ut
= ultimate tensile strength, s
uc
= ultimate compressive strength.
• FOS :
FOS =
s
ut
s
1
(tension) or FOS =
s
uc
|s
3
|
(compression)
1
• Coulomb-Mohr Theory :
s
1
s
ut
-
s
3
s
uc
= 1
• Modified Mohr Theory : A djusts Coulomb-Mohr for better accur acy , often
gr aphical or emp irical for complex stress states.
Principal Stresses Calculation
• 2D Stres s State :
s
1,2
=
s
x
+s
y
2
±
v
(
s
x
-s
y
2
)
2
+t
2
xy
• Maximum Shea r Stress :
t
max
=
v
(
s
x
-s
y
2
)
2
+t
2
xy
=
s
1
-s
2
2
• Principal An gle :
tan2?
p
=
2t
xy
s
x
-s
y
where ?
p
= angle of principal plane.
Applications and Notes
• von Mises : Preferred for ductile materials, most accur ate for predicting
yielding under complex loading.
• Tresca : Conservative, simpler to apply , used for quick design checks.
• Maximum Normal Stress : Suitable for brittle materials where fr acture
occurs due to ma ximum tensile stress.
• Coulomb-Mohr : Used for brittle materials with different tensile and com-
pressive stre ngths.
• F actor of Safety : Typically applied to ensure safety , e.g., s
y
/ FOS for allow-
able stress .
Str ain Energy Based Theories
• Distortion Energy (von Mises) :
U
d
=
1+?
3E
[
(s
1
-s
2
)
2
+(s
2
-s
3
)
2
+(s
3
-s
1
)
2
]
where U
d
= distortion energy , ? = Pois son’ s r atio, E = Y oung’ s modulus.
2
Page 3


F ormula Sheet: Theories of F ailure
Introduction to Theories of F ailure
• Definition : Theories of failure predict the conditions under which a mate-
rial fails under complex loading, used to design components safely .
• Applications : Primarily for ductile (e.g., steel) and brittle (e.g., cast iron)
materials under static or dynamic loading.
• Principal Stresses : F or a 3D stress state, principal stresses (s
1
,s
2
,s
3
) are
calculated, t ypically ordered as s
1
= s
2
= s
3
.
F ailure Theories for Ductile Materials
• Maximum Shea r Stress Theory (Tresca) :
s
1
-s
3
= s
y
where s
y
= yield strength.
• F actor o f Safety (FOS) :
FOS =
s
y
s
1
-s
3
• von Mises (Distortion Energy) Theory :
v
(s
1
-s
2
)
2
+(s
2
-s
3
)
2
+(s
3
-s
1
)
2
2
= s
y
or
s
von Mis es
=
v
s
2
x
+s
2
y
-s
x
s
y
+3t
2
xy
= s
y
where s
x
,s
y
= normal stresses, t
xy
= shear stress.
• FOS for v on Mises :
FOS =
s
y
s
von Mises
F ailure Theories for Brittle Materials
• Maximum Nor mal Stress (Rankine) Theory :
s
1
= s
ut
or |s
3
|= s
uc
where s
ut
= ultimate tensile strength, s
uc
= ultimate compressive strength.
• FOS :
FOS =
s
ut
s
1
(tension) or FOS =
s
uc
|s
3
|
(compression)
1
• Coulomb-Mohr Theory :
s
1
s
ut
-
s
3
s
uc
= 1
• Modified Mohr Theory : A djusts Coulomb-Mohr for better accur acy , often
gr aphical or emp irical for complex stress states.
Principal Stresses Calculation
• 2D Stres s State :
s
1,2
=
s
x
+s
y
2
±
v
(
s
x
-s
y
2
)
2
+t
2
xy
• Maximum Shea r Stress :
t
max
=
v
(
s
x
-s
y
2
)
2
+t
2
xy
=
s
1
-s
2
2
• Principal An gle :
tan2?
p
=
2t
xy
s
x
-s
y
where ?
p
= angle of principal plane.
Applications and Notes
• von Mises : Preferred for ductile materials, most accur ate for predicting
yielding under complex loading.
• Tresca : Conservative, simpler to apply , used for quick design checks.
• Maximum Normal Stress : Suitable for brittle materials where fr acture
occurs due to ma ximum tensile stress.
• Coulomb-Mohr : Used for brittle materials with different tensile and com-
pressive stre ngths.
• F actor of Safety : Typically applied to ensure safety , e.g., s
y
/ FOS for allow-
able stress .
Str ain Energy Based Theories
• Distortion Energy (von Mises) :
U
d
=
1+?
3E
[
(s
1
-s
2
)
2
+(s
2
-s
3
)
2
+(s
3
-s
1
)
2
]
where U
d
= distortion energy , ? = Pois son’ s r atio, E = Y oung’ s modulus.
2
• T otal Str ain Energy :
U =
1
2E
[
s
2
1
+s
2
2
+s
2
3
-2?(s
1
s
2
+s
2
s
3
+s
3
s
1
)
]
3