Formula Sheet: Strain Energy | Solid Mechanics - Mechanical Engineering PDF Download

 Page 1


F ormula Sheet: Str ain Energy
Introduction to Str ain Energy
• Definition : Str ain energy is the energy stored in a body due to elastic de-
formation under applied loads.
• Assumptions : Linear elastic material, small deformations, Hook e’ s law ap-
plies.
• Str ain Ener gy Density :
u=
1
2
s?
whereu = str ain energy per unit volume,s = stress,? = str ain.
Str ain Energy for Different Loading Conditions
• Axial Loading :
U =
1
2
?
P
2
EA
dx
where U = str ain energy , P = axial load, E = Y oung’ s modulus, A = cross-
sectional a rea.
• F or Const ant Axial Load :
U =
P
2
L
2EA
whereL = length.
• Bending (Fle xur al) Loading :
U =
?
M
2
2EI
dx
whereM = bending moment,I = moment of inertia.
• F or Const ant Bending Moment :
U =
M
2
L
2EI
• T orsional L oading :
U =
?
T
2
2GJ
dx
whereT = torque,G = shear modulus,J = polar moment of inertia.
• F or Const ant T orque :
U =
T
2
L
2GJ
1
Page 2


F ormula Sheet: Str ain Energy
Introduction to Str ain Energy
• Definition : Str ain energy is the energy stored in a body due to elastic de-
formation under applied loads.
• Assumptions : Linear elastic material, small deformations, Hook e’ s law ap-
plies.
• Str ain Ener gy Density :
u=
1
2
s?
whereu = str ain energy per unit volume,s = stress,? = str ain.
Str ain Energy for Different Loading Conditions
• Axial Loading :
U =
1
2
?
P
2
EA
dx
where U = str ain energy , P = axial load, E = Y oung’ s modulus, A = cross-
sectional a rea.
• F or Const ant Axial Load :
U =
P
2
L
2EA
whereL = length.
• Bending (Fle xur al) Loading :
U =
?
M
2
2EI
dx
whereM = bending moment,I = moment of inertia.
• F or Const ant Bending Moment :
U =
M
2
L
2EI
• T orsional L oading :
U =
?
T
2
2GJ
dx
whereT = torque,G = shear modulus,J = polar moment of inertia.
• F or Const ant T orque :
U =
T
2
L
2GJ
1
• Shear Loa ding :
U =
?
V
2
2GA
dx
whereV = shear force,A = cros s-sectional area,G = shear modulus.
• F or Const ant Shear F orce (Rectangular Section) :
U =
6V
2
L
5GA
(Note: F actor
6
5
accounts for shear stress distribution.)
Castigliano’ s Theorem
• First Theo rem (Deflection) :
d
i
=
?U
?P
i
whered
i
= deflection at point of application of load P
i
, U = total str ain en-
ergy .
• Second Theo rem (Slope or Rotation) :
?
i
=
?U
?M
i
where?
i
= rotation at point of application of momentM
i
.
• Application : Used to calculate deflections or slopes b y differentiating str ain
energy with resp ect to applied loads or moments.
Str ain Energy for Combined Loading
• T otal Str ain Energy :
U
total
=U
axial
+U
bending
+U
torsion
+U
shear
• Example (Co mbined Axial and Bending) :
U =
?
P
2
2EA
dx+
?
M
2
2EI
dx
W ork-Energy Relationship
• W ork Done b y External Loads = Str ain energy s tored:
W =U
• F or Linear Elastic S ystems :
W =
1
2
Pd (for axial load)
W =
1
2
M? (for bending moment)
2
Page 3


F ormula Sheet: Str ain Energy
Introduction to Str ain Energy
• Definition : Str ain energy is the energy stored in a body due to elastic de-
formation under applied loads.
• Assumptions : Linear elastic material, small deformations, Hook e’ s law ap-
plies.
• Str ain Ener gy Density :
u=
1
2
s?
whereu = str ain energy per unit volume,s = stress,? = str ain.
Str ain Energy for Different Loading Conditions
• Axial Loading :
U =
1
2
?
P
2
EA
dx
where U = str ain energy , P = axial load, E = Y oung’ s modulus, A = cross-
sectional a rea.
• F or Const ant Axial Load :
U =
P
2
L
2EA
whereL = length.
• Bending (Fle xur al) Loading :
U =
?
M
2
2EI
dx
whereM = bending moment,I = moment of inertia.
• F or Const ant Bending Moment :
U =
M
2
L
2EI
• T orsional L oading :
U =
?
T
2
2GJ
dx
whereT = torque,G = shear modulus,J = polar moment of inertia.
• F or Const ant T orque :
U =
T
2
L
2GJ
1
• Shear Loa ding :
U =
?
V
2
2GA
dx
whereV = shear force,A = cros s-sectional area,G = shear modulus.
• F or Const ant Shear F orce (Rectangular Section) :
U =
6V
2
L
5GA
(Note: F actor
6
5
accounts for shear stress distribution.)
Castigliano’ s Theorem
• First Theo rem (Deflection) :
d
i
=
?U
?P
i
whered
i
= deflection at point of application of load P
i
, U = total str ain en-
ergy .
• Second Theo rem (Slope or Rotation) :
?
i
=
?U
?M
i
where?
i
= rotation at point of application of momentM
i
.
• Application : Used to calculate deflections or slopes b y differentiating str ain
energy with resp ect to applied loads or moments.
Str ain Energy for Combined Loading
• T otal Str ain Energy :
U
total
=U
axial
+U
bending
+U
torsion
+U
shear
• Example (Co mbined Axial and Bending) :
U =
?
P
2
2EA
dx+
?
M
2
2EI
dx
W ork-Energy Relationship
• W ork Done b y External Loads = Str ain energy s tored:
W =U
• F or Linear Elastic S ystems :
W =
1
2
Pd (for axial load)
W =
1
2
M? (for bending moment)
2
Resilience
• Modulus of Resilience (Str ain energy per unit volume at yield point):
u
r
=
1
2
s
y
?
y
=
s
2
y
2E
wheres
y
= yield stress,?
y
= yield str ain.
• T oughness : T otal str ain energy per unit volume until fr acture (area under
stress-str ain curve).
3