PPT: Strain Energy & Castigliano’s Theorem | Structural Analysis - Civil Engineering (CE) PDF Download

 Page 1


Strain Strain Energy and Energy and Castigliano’s Castigliano’s Theorem Theorem
Page 2


Strain Strain Energy and Energy and Castigliano’s Castigliano’s Theorem Theorem
o A uniform rod is subjected to a slowly increasing load
o The elementary work done by the load P as the rod
elongates by a small dx is
which is equal to the area of width dx under the
load-deformation diagram.
dU Pdx elementary work ==
o The total work done by the load for deformation x
1
,
which results in an increase of strain energy in the rod.
1
0
x
U P dx total work strain energy = ==
ò
1
2
11
1 11 22
0
x
U kx dx kx Px = ==
ò
o In the case of a linear elastic deformation,
Strain Energy Strain Energy
Page 3


Strain Strain Energy and Energy and Castigliano’s Castigliano’s Theorem Theorem
o A uniform rod is subjected to a slowly increasing load
o The elementary work done by the load P as the rod
elongates by a small dx is
which is equal to the area of width dx under the
load-deformation diagram.
dU Pdx elementary work ==
o The total work done by the load for deformation x
1
,
which results in an increase of strain energy in the rod.
1
0
x
U P dx total work strain energy = ==
ò
1
2
11
1 11 22
0
x
U kx dx kx Px = ==
ò
o In the case of a linear elastic deformation,
Strain Energy Strain Energy Strain Energy Density Strain Energy Density
o To eliminate the effects of size, evaluate the strain
energy per unit volume,
1
1
0
0
x
xx
U P dx
V AL
u d strain energy density
e
se
=
==
ò
ò
o As the material is unloaded, the stress returns to zero but there is a
permanent deformation. Only the strain energy represented by the
triangular area is recovered.
o Remainder of the energy spent in deforming the material is dissipated as
heat.
o The total strain energy density is equal to the area under the curve to e
1
.
Page 4


Strain Strain Energy and Energy and Castigliano’s Castigliano’s Theorem Theorem
o A uniform rod is subjected to a slowly increasing load
o The elementary work done by the load P as the rod
elongates by a small dx is
which is equal to the area of width dx under the
load-deformation diagram.
dU Pdx elementary work ==
o The total work done by the load for deformation x
1
,
which results in an increase of strain energy in the rod.
1
0
x
U P dx total work strain energy = ==
ò
1
2
11
1 11 22
0
x
U kx dx kx Px = ==
ò
o In the case of a linear elastic deformation,
Strain Energy Strain Energy Strain Energy Density Strain Energy Density
o To eliminate the effects of size, evaluate the strain
energy per unit volume,
1
1
0
0
x
xx
U P dx
V AL
u d strain energy density
e
se
=
==
ò
ò
o As the material is unloaded, the stress returns to zero but there is a
permanent deformation. Only the strain energy represented by the
triangular area is recovered.
o Remainder of the energy spent in deforming the material is dissipated as
heat.
o The total strain energy density is equal to the area under the curve to e
1
.
Strain Strain- -Energy Density Energy Density
o The strain energy density resulting from
setting e
1
= e
R
is the modulus of toughness.
o If the stress remains within the proportional
limit,
1
22
11
0
22
xx
E
u Ed
E
e
es
ee = ==
ò
o The strain energy density resulting from
setting s
1
= s
Y
is the modulus of resilience.
2
2
Y
Y
u modulus of resilience
E
s
==
Page 5


Strain Strain Energy and Energy and Castigliano’s Castigliano’s Theorem Theorem
o A uniform rod is subjected to a slowly increasing load
o The elementary work done by the load P as the rod
elongates by a small dx is
which is equal to the area of width dx under the
load-deformation diagram.
dU Pdx elementary work ==
o The total work done by the load for deformation x
1
,
which results in an increase of strain energy in the rod.
1
0
x
U P dx total work strain energy = ==
ò
1
2
11
1 11 22
0
x
U kx dx kx Px = ==
ò
o In the case of a linear elastic deformation,
Strain Energy Strain Energy Strain Energy Density Strain Energy Density
o To eliminate the effects of size, evaluate the strain
energy per unit volume,
1
1
0
0
x
xx
U P dx
V AL
u d strain energy density
e
se
=
==
ò
ò
o As the material is unloaded, the stress returns to zero but there is a
permanent deformation. Only the strain energy represented by the
triangular area is recovered.
o Remainder of the energy spent in deforming the material is dissipated as
heat.
o The total strain energy density is equal to the area under the curve to e
1
.
Strain Strain- -Energy Density Energy Density
o The strain energy density resulting from
setting e
1
= e
R
is the modulus of toughness.
o If the stress remains within the proportional
limit,
1
22
11
0
22
xx
E
u Ed
E
e
es
ee = ==
ò
o The strain energy density resulting from
setting s
1
= s
Y
is the modulus of resilience.
2
2
Y
Y
u modulus of resilience
E
s
==
Elastic Strain Energy for Normal Stresses Elastic Strain Energy for Normal Stresses
o In an element with a nonuniform stress distribution,
0
lim total strain energy
V
U dU
u U u dV
V dV
D®
D
= = ==
D
ò
o For values of u < u
Y
, i.e., below the proportional
limit,
2
2
x
U dV elastic strain energy
E
s
==
ò
o Under axial loading,
x
P A dV A dx s==
2
0
2
L
P
U dx
AE
=
ò
2
2
PL
U
AE
=
o For a rod of uniform cross section,