Formula Sheets: Stress and Strain | Strength of Materials (SOM) - Mechanical Engineering PDF Download

 Page 1


Stress and Strain F orm ula Sheet for Mec hanical
GA TE
Stress
• Normal Stress : F orce p er unit area p erp endicular to the surface.
s =
F
A
(Units: Pa or Nm
-2
)
where F is the normal force, A is the cross-sectional area.
• Shear Stress : F orce p er unit area parallel to the surface.
t =
F
A
where F is the shear force.
• A v erage Normal Stress : F or v arying stress distribution.
s
a vg
=
P
A
where P is the resultan t force.
Strain
• Normal Strain : Deformation p er unit length along the axial direction.
? =
?L
L
where ?L is the c hange in length, L is the original length (dimensionless).
• Shear Strain : Angular deformation due to shear stress.
? =tan?˜? (for small angles, in radians)
where ? is the angle of deformation.
• V olumetric Strain : Change in v olume p er unit v olume.
?
v
=
?V
V
=?
x
+?
y
+?
z
where ?V is the c hange in v olume, V is the original v olume.
Stress-Strain Relationships
• Ho ok e’s La w (Uniaxial) : Linear relationship within elastic limit.
s =E?
where E is Y oung’s mo dulus (Pa ).
1
Page 2


Stress and Strain F orm ula Sheet for Mec hanical
GA TE
Stress
• Normal Stress : F orce p er unit area p erp endicular to the surface.
s =
F
A
(Units: Pa or Nm
-2
)
where F is the normal force, A is the cross-sectional area.
• Shear Stress : F orce p er unit area parallel to the surface.
t =
F
A
where F is the shear force.
• A v erage Normal Stress : F or v arying stress distribution.
s
a vg
=
P
A
where P is the resultan t force.
Strain
• Normal Strain : Deformation p er unit length along the axial direction.
? =
?L
L
where ?L is the c hange in length, L is the original length (dimensionless).
• Shear Strain : Angular deformation due to shear stress.
? =tan?˜? (for small angles, in radians)
where ? is the angle of deformation.
• V olumetric Strain : Change in v olume p er unit v olume.
?
v
=
?V
V
=?
x
+?
y
+?
z
where ?V is the c hange in v olume, V is the original v olume.
Stress-Strain Relationships
• Ho ok e’s La w (Uniaxial) : Linear relationship within elastic limit.
s =E?
where E is Y oung’s mo dulus (Pa ).
1
• Shear Stress-Strain : Shear stress prop ortional to shear strain.
t =G?
where G is the shear mo dulus (Pa ).
• P oisson’s Ratio : Ratio of lateral strain to axial strain.
? =-
?
lateral
?
axial
T ypically , 0=?= 0.5 for most materials.
• Generalized Ho ok e’s La w (3D) : Stress-strain relationship in three dimensions.
?
x
=
1
E
[s
x
-?(s
y
+s
z
)]
?
y
=
1
E
[s
y
-?(s
x
+s
z
)]
?
z
=
1
E
[s
z
-?(s
x
+s
y
)]
• Shear Strain in 3D :
?
xy
=
t
xy
G
, ?
yz
=
t
yz
G
, ?
zx
=
t
zx
G
Elastic Constan ts
• Y oung’s Mo dulus (E ) : Measure of stiffness.
E =
s
?
• Shear Mo dulus (G ) : Resistance to shear deformation.
G =
t
?
• Bulk Mo dulus (K ) : Resistance to uniform compression.
K =-
?P
?V/V
where ?P is the c hange in pressure.
• Relationship b et w een Elastic Constan ts :
E =2G(1+?), E =3K(1-2?), E =
9KG
3K +G
2
Page 3


Stress and Strain F orm ula Sheet for Mec hanical
GA TE
Stress
• Normal Stress : F orce p er unit area p erp endicular to the surface.
s =
F
A
(Units: Pa or Nm
-2
)
where F is the normal force, A is the cross-sectional area.
• Shear Stress : F orce p er unit area parallel to the surface.
t =
F
A
where F is the shear force.
• A v erage Normal Stress : F or v arying stress distribution.
s
a vg
=
P
A
where P is the resultan t force.
Strain
• Normal Strain : Deformation p er unit length along the axial direction.
? =
?L
L
where ?L is the c hange in length, L is the original length (dimensionless).
• Shear Strain : Angular deformation due to shear stress.
? =tan?˜? (for small angles, in radians)
where ? is the angle of deformation.
• V olumetric Strain : Change in v olume p er unit v olume.
?
v
=
?V
V
=?
x
+?
y
+?
z
where ?V is the c hange in v olume, V is the original v olume.
Stress-Strain Relationships
• Ho ok e’s La w (Uniaxial) : Linear relationship within elastic limit.
s =E?
where E is Y oung’s mo dulus (Pa ).
1
• Shear Stress-Strain : Shear stress prop ortional to shear strain.
t =G?
where G is the shear mo dulus (Pa ).
• P oisson’s Ratio : Ratio of lateral strain to axial strain.
? =-
?
lateral
?
axial
T ypically , 0=?= 0.5 for most materials.
• Generalized Ho ok e’s La w (3D) : Stress-strain relationship in three dimensions.
?
x
=
1
E
[s
x
-?(s
y
+s
z
)]
?
y
=
1
E
[s
y
-?(s
x
+s
z
)]
?
z
=
1
E
[s
z
-?(s
x
+s
y
)]
• Shear Strain in 3D :
?
xy
=
t
xy
G
, ?
yz
=
t
yz
G
, ?
zx
=
t
zx
G
Elastic Constan ts
• Y oung’s Mo dulus (E ) : Measure of stiffness.
E =
s
?
• Shear Mo dulus (G ) : Resistance to shear deformation.
G =
t
?
• Bulk Mo dulus (K ) : Resistance to uniform compression.
K =-
?P
?V/V
where ?P is the c hange in pressure.
• Relationship b et w een Elastic Constan ts :
E =2G(1+?), E =3K(1-2?), E =
9KG
3K +G
2
Thermal Strain
• Thermal Strain : Strain due to temp erature c hange.
?
T
=a?T
where a is the co e?icien t of thermal expansion ( K
-1
), ?T is the temp erature c hange
(K ).
• Thermal Stress (if constrained):
s
T
=Ea?T
Key Notes
• T ypical v alues: E
steel
˜ 200GPa , ?
steel
˜ 0.3 , G
steel
˜ 77GPa .
• Use SI units for GA TE calculations.
• F or 3D stress states, c hec k for plane stress (s
z
=0 ) or plane strain (?
z
=0 ) conditions.
• Ensure p ositiv e tensile stress and negativ e compressiv e stress con v en tions are follo w ed.
3