Formula Sheets: Principal Stresses & Strains (Mohr's Circle) | Solid Mechanics - Mechanical Engineering PDF Download

 Page 1


Principal Str ess and Strain F orm ula Sheet for
Mec hanical GA TE
Principal Stresses
• Principal Stresses (2D) : Maxim um and minim um normal stresses in a plane stress
state.
s
1,2
=
s
x
+s
y
2
±
v
(
s
x
-s
y
2
)
2
+t
2
xy
where s
x
,s
y
are normal stresses, t
xy
is shear stress (Pa ).
• Orien tation of Principal Planes : Angle of principal planes from the x-axis.
tan2?
p
=
2t
xy
s
x
-s
y
where ?
p
is the angle to the principal plane (in radians or degrees).
• Maxim um Shear Stress (2D) : Occurs at 45° to p rincipal planes.
t
max
=
v
(
s
x
-s
y
2
)
2
+t
2
xy
=
s
1
-s
2
2
• A v erage Normal Stress : Constan t across all planes.
s
a vg
=
s
x
+s
y
2
Principal Strains
• Principal Strains (2D) : Maxim um and minim um normal strains in a plane strain
state.
?
1,2
=
?
x
+?
y
2
±
v
(
?
x
-?
y
2
)
2
+
(
?
xy
2
)
2
where ?
x
,?
y
are normal strains, ?
xy
is shear strain (dimensionless).
• Orien tation of Principal Strains : Angle of principal strain planes.
tan2?
p
=
?
xy
?
x
-?
y
where ?
p
is the angle to the principal strain plane.
• Maxim um Shear Strain :
?
max
=
v
(?
x
-?
y
)
2
+?
2
xy
= ?
1
-?
2
1
Page 2


Principal Str ess and Strain F orm ula Sheet for
Mec hanical GA TE
Principal Stresses
• Principal Stresses (2D) : Maxim um and minim um normal stresses in a plane stress
state.
s
1,2
=
s
x
+s
y
2
±
v
(
s
x
-s
y
2
)
2
+t
2
xy
where s
x
,s
y
are normal stresses, t
xy
is shear stress (Pa ).
• Orien tation of Principal Planes : Angle of principal planes from the x-axis.
tan2?
p
=
2t
xy
s
x
-s
y
where ?
p
is the angle to the principal plane (in radians or degrees).
• Maxim um Shear Stress (2D) : Occurs at 45° to p rincipal planes.
t
max
=
v
(
s
x
-s
y
2
)
2
+t
2
xy
=
s
1
-s
2
2
• A v erage Normal Stress : Constan t across all planes.
s
a vg
=
s
x
+s
y
2
Principal Strains
• Principal Strains (2D) : Maxim um and minim um normal strains in a plane strain
state.
?
1,2
=
?
x
+?
y
2
±
v
(
?
x
-?
y
2
)
2
+
(
?
xy
2
)
2
where ?
x
,?
y
are normal strains, ?
xy
is shear strain (dimensionless).
• Orien tation of Principal Strains : Angle of principal strain planes.
tan2?
p
=
?
xy
?
x
-?
y
where ?
p
is the angle to the principal strain plane.
• Maxim um Shear Strain :
?
max
=
v
(?
x
-?
y
)
2
+?
2
xy
= ?
1
-?
2
1
Stress T ransformation
• Normal Stress at Angle ? : Stress on a plane rotated b y angle ? .
s
?
=
s
x
+s
y
2
+
s
x
-s
y
2
cos2? +t
xy
sin2?
• Shear Stress at Angle ? : Shear stress on the rotated plane.
t
?
=-
s
x
-s
y
2
sin2? +t
xy
cos2?
• Plane Stress Condition : s
z
= t
xz
= t
yz
= 0 .
Mohr’s Circle
• Cen ter of Mohr’s Circle (Stress) :
Cen ter =
(
s
x
+s
y
2
,0
)
• Radius of Mohr’s Circle (Stress) :
R =
v
(
s
x
-s
y
2
)
2
+t
2
xy
• Principal Stresses from Mohr’s Circle :
s
1,2
=
s
x
+s
y
2
±R
• Mohr’s Circle for Strain : Similar to stress, replace s
x
,s
y
,t
xy
with ?
x
,?
y
,
?xy
2
.
Principal Stresses (3D)
• Principal Stresses : Obtained b y solving the c haracteristic equation of the stress
tensor.
s
3
-I
1
s
2
+I
2
s-I
3
= 0
where:
I
1
= s
x
+s
y
+s
z
I
2
= s
x
s
y
+s
y
s
z
+s
z
s
x
-t
2
xy
-t
2
yz
-t
2
zx
I
3
= s
x
s
y
s
z
+2t
xy
t
yz
t
zx
-s
x
t
2
yz
-s
y
t
2
zx
-s
z
t
2
xy
• Maxim um Shear Stress (3D) :
t
max
=
s
max
-s
min
2
2
Page 3


Principal Str ess and Strain F orm ula Sheet for
Mec hanical GA TE
Principal Stresses
• Principal Stresses (2D) : Maxim um and minim um normal stresses in a plane stress
state.
s
1,2
=
s
x
+s
y
2
±
v
(
s
x
-s
y
2
)
2
+t
2
xy
where s
x
,s
y
are normal stresses, t
xy
is shear stress (Pa ).
• Orien tation of Principal Planes : Angle of principal planes from the x-axis.
tan2?
p
=
2t
xy
s
x
-s
y
where ?
p
is the angle to the principal plane (in radians or degrees).
• Maxim um Shear Stress (2D) : Occurs at 45° to p rincipal planes.
t
max
=
v
(
s
x
-s
y
2
)
2
+t
2
xy
=
s
1
-s
2
2
• A v erage Normal Stress : Constan t across all planes.
s
a vg
=
s
x
+s
y
2
Principal Strains
• Principal Strains (2D) : Maxim um and minim um normal strains in a plane strain
state.
?
1,2
=
?
x
+?
y
2
±
v
(
?
x
-?
y
2
)
2
+
(
?
xy
2
)
2
where ?
x
,?
y
are normal strains, ?
xy
is shear strain (dimensionless).
• Orien tation of Principal Strains : Angle of principal strain planes.
tan2?
p
=
?
xy
?
x
-?
y
where ?
p
is the angle to the principal strain plane.
• Maxim um Shear Strain :
?
max
=
v
(?
x
-?
y
)
2
+?
2
xy
= ?
1
-?
2
1
Stress T ransformation
• Normal Stress at Angle ? : Stress on a plane rotated b y angle ? .
s
?
=
s
x
+s
y
2
+
s
x
-s
y
2
cos2? +t
xy
sin2?
• Shear Stress at Angle ? : Shear stress on the rotated plane.
t
?
=-
s
x
-s
y
2
sin2? +t
xy
cos2?
• Plane Stress Condition : s
z
= t
xz
= t
yz
= 0 .
Mohr’s Circle
• Cen ter of Mohr’s Circle (Stress) :
Cen ter =
(
s
x
+s
y
2
,0
)
• Radius of Mohr’s Circle (Stress) :
R =
v
(
s
x
-s
y
2
)
2
+t
2
xy
• Principal Stresses from Mohr’s Circle :
s
1,2
=
s
x
+s
y
2
±R
• Mohr’s Circle for Strain : Similar to stress, replace s
x
,s
y
,t
xy
with ?
x
,?
y
,
?xy
2
.
Principal Stresses (3D)
• Principal Stresses : Obtained b y solving the c haracteristic equation of the stress
tensor.
s
3
-I
1
s
2
+I
2
s-I
3
= 0
where:
I
1
= s
x
+s
y
+s
z
I
2
= s
x
s
y
+s
y
s
z
+s
z
s
x
-t
2
xy
-t
2
yz
-t
2
zx
I
3
= s
x
s
y
s
z
+2t
xy
t
yz
t
zx
-s
x
t
2
yz
-s
y
t
2
zx
-s
z
t
2
xy
• Maxim um Shear Stress (3D) :
t
max
=
s
max
-s
min
2
2
Stress-Strain Relationship
• Ho ok e’s La w for Principal Strains (3D) :
?
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
)]
where E is Y oung’s mo dulus (Pa ), ? is P oisson’s ratio.
• Shear Strain :
?
xy
=
t
xy
G
, G =
E
2(1+?)
Key Notes
• Use SI units: stresses in Pa , strains are dimensionless.
• T ypical v alues: E
steel
˜ 200GPa , ?
steel
˜ 0.3 , G
steel
˜ 77GPa .
• F or Mohr’s circle, plot (s
x
,t
xy
) and (s
y
,-t
xy
) for 2D stress state.
• In plane stress, principal stresses are in the plane; third principal stress is zero.
• Chec k sign con v en tions: tensile stress is p ositiv e, compressiv e stress is negativ e.
3