Short Notes: Energy Methods | Short Notes for Civil Engineering - Civil Engineering (CE) PDF Download

 Page 1


Energy Methods
Strain Energy: The work done by the load in straining the body is stored within the 
strained material in the form of strain energy.
Strain energy,
U =
P-L 
2 AE
Put
Or
P =
AEAl
L
V
Strain energy diagram
Page 2


Energy Methods
Strain Energy: The work done by the load in straining the body is stored within the 
strained material in the form of strain energy.
Strain energy,
U =
P-L 
2 AE
Put
Or
P =
AEAl
L
V
Strain energy diagram
Proof Resilience: The maximum strain energy that can be stored in a material is 
known as proof resilience.
1 P-L 
u = — x
V 2AE
u =
a
Where,
p 1 - 2
a = —.u= — Ee 
A 2
Strain energy of prismatic bar with varying sections:
P:L
u =:
_ p 2 
2 E
2 AE
l : _ l 2
A, A,
k \
T
ti
V r +
I
Prismatic bar
Strain energy of non-prismatic bar with varying axial force:
, 2 p -
U = f-
J o 7
° 2 £.1
£ k
Ay = Cross-section of differential section.
w
, .
i
dx
T
¦
-
Non-prismatic b a r
Stresses due to
1. Gradual Loading:-
2. Sudden Loading:-
3. Impact Loading:- Work done by falling weigth P is
Page 3


Energy Methods
Strain Energy: The work done by the load in straining the body is stored within the 
strained material in the form of strain energy.
Strain energy,
U =
P-L 
2 AE
Put
Or
P =
AEAl
L
V
Strain energy diagram
Proof Resilience: The maximum strain energy that can be stored in a material is 
known as proof resilience.
1 P-L 
u = — x
V 2AE
u =
a
Where,
p 1 - 2
a = —.u= — Ee 
A 2
Strain energy of prismatic bar with varying sections:
P:L
u =:
_ p 2 
2 E
2 AE
l : _ l 2
A, A,
k \
T
ti
V r +
I
Prismatic bar
Strain energy of non-prismatic bar with varying axial force:
, 2 p -
U = f-
J o 7
° 2 £.1
£ k
Ay = Cross-section of differential section.
w
, .
i
dx
T
¦
-
Non-prismatic b a r
Stresses due to
1. Gradual Loading:-
2. Sudden Loading:-
3. Impact Loading:- Work done by falling weigth P is
l — 1
t e u a g u r
* - Vertical Bar
L HI
iff- Load (P)
T
h
l
........................ XA
r
Work = P(h + A ) = i / h +
.....(a)
Work stored in the bar
Work = — x AL
2E
.(b)
By equating, stress will be
I f A
and
PL
st
AE
A = A ,( + V ( A s < f + 2 h A st 
if h is very small then A « yj‘ 2h A s < 
Strain Energy in Torsion:
U = —T O = U l L 
2 4 C J
For solid shaft,
r 2
U = ------X V olum e o f sh a ft
4(7
For hollow shaft,
r‘ 2 ( D 2 + d 2
f / = - L .
4 (7 '
x V olum e of sh a ft
D
Castigliano's First Theorem: It the strain energy of an elastic structure can be 
expressed as a function of generalized displacement, then the partial derivative of 
the strain energy with respect to generalized displacement gives the generalized 
force
Page 4


Energy Methods
Strain Energy: The work done by the load in straining the body is stored within the 
strained material in the form of strain energy.
Strain energy,
U =
P-L 
2 AE
Put
Or
P =
AEAl
L
V
Strain energy diagram
Proof Resilience: The maximum strain energy that can be stored in a material is 
known as proof resilience.
1 P-L 
u = — x
V 2AE
u =
a
Where,
p 1 - 2
a = —.u= — Ee 
A 2
Strain energy of prismatic bar with varying sections:
P:L
u =:
_ p 2 
2 E
2 AE
l : _ l 2
A, A,
k \
T
ti
V r +
I
Prismatic bar
Strain energy of non-prismatic bar with varying axial force:
, 2 p -
U = f-
J o 7
° 2 £.1
£ k
Ay = Cross-section of differential section.
w
, .
i
dx
T
¦
-
Non-prismatic b a r
Stresses due to
1. Gradual Loading:-
2. Sudden Loading:-
3. Impact Loading:- Work done by falling weigth P is
l — 1
t e u a g u r
* - Vertical Bar
L HI
iff- Load (P)
T
h
l
........................ XA
r
Work = P(h + A ) = i / h +
.....(a)
Work stored in the bar
Work = — x AL
2E
.(b)
By equating, stress will be
I f A
and
PL
st
AE
A = A ,( + V ( A s < f + 2 h A st 
if h is very small then A « yj‘ 2h A s < 
Strain Energy in Torsion:
U = —T O = U l L 
2 4 C J
For solid shaft,
r 2
U = ------X V olum e o f sh a ft
4(7
For hollow shaft,
r‘ 2 ( D 2 + d 2
f / = - L .
4 (7 '
x V olum e of sh a ft
D
Castigliano's First Theorem: It the strain energy of an elastic structure can be 
expressed as a function of generalized displacement, then the partial derivative of 
the strain energy with respect to generalized displacement gives the generalized 
force
u =
r lM'dx 
2El
Slope:
$ -
6U
6 J 1 \
6M rfc 
dTT] El
9=
6U 
6M,
dx
El
Theories of Failure: Theories of failure are defined as following groups. 
Maximum Principal Stress Theory (Rankine theory):
• According to this theory, permanent set takes place under a state of complex 
stress, when the value of maximum principal stress is equal to that of yield 
point stress as found in a simple tensile test.
• For design, critical maximum principal stress (oi) must not exceed the 
working stress (
< r y
) for the material.
ff! < (T y
Note: For bittle material, it gives satisfactory result. Yield criteria for 3D stress 
system,
a , = a v or a } = a[
Where, oy = Yield stress point in simple tension, and oy = Yield stress point in 
simple compression.
Stress&s on 
rectangular section
Maximum Principal Strain Theory (St. Venant's theory):
According to this theory, a ductile material begins to yield when the maximum 
principal strain at which yielding occurs in simple tension.
For 3D stress system,
ej,2,3< h r
a, 1 , N
h = —~ — (a\ + o -3 )
• m•
Page 5


Energy Methods
Strain Energy: The work done by the load in straining the body is stored within the 
strained material in the form of strain energy.
Strain energy,
U =
P-L 
2 AE
Put
Or
P =
AEAl
L
V
Strain energy diagram
Proof Resilience: The maximum strain energy that can be stored in a material is 
known as proof resilience.
1 P-L 
u = — x
V 2AE
u =
a
Where,
p 1 - 2
a = —.u= — Ee 
A 2
Strain energy of prismatic bar with varying sections:
P:L
u =:
_ p 2 
2 E
2 AE
l : _ l 2
A, A,
k \
T
ti
V r +
I
Prismatic bar
Strain energy of non-prismatic bar with varying axial force:
, 2 p -
U = f-
J o 7
° 2 £.1
£ k
Ay = Cross-section of differential section.
w
, .
i
dx
T
¦
-
Non-prismatic b a r
Stresses due to
1. Gradual Loading:-
2. Sudden Loading:-
3. Impact Loading:- Work done by falling weigth P is
l — 1
t e u a g u r
* - Vertical Bar
L HI
iff- Load (P)
T
h
l
........................ XA
r
Work = P(h + A ) = i / h +
.....(a)
Work stored in the bar
Work = — x AL
2E
.(b)
By equating, stress will be
I f A
and
PL
st
AE
A = A ,( + V ( A s < f + 2 h A st 
if h is very small then A « yj‘ 2h A s < 
Strain Energy in Torsion:
U = —T O = U l L 
2 4 C J
For solid shaft,
r 2
U = ------X V olum e o f sh a ft
4(7
For hollow shaft,
r‘ 2 ( D 2 + d 2
f / = - L .
4 (7 '
x V olum e of sh a ft
D
Castigliano's First Theorem: It the strain energy of an elastic structure can be 
expressed as a function of generalized displacement, then the partial derivative of 
the strain energy with respect to generalized displacement gives the generalized 
force
u =
r lM'dx 
2El
Slope:
$ -
6U
6 J 1 \
6M rfc 
dTT] El
9=
6U 
6M,
dx
El
Theories of Failure: Theories of failure are defined as following groups. 
Maximum Principal Stress Theory (Rankine theory):
• According to this theory, permanent set takes place under a state of complex 
stress, when the value of maximum principal stress is equal to that of yield 
point stress as found in a simple tensile test.
• For design, critical maximum principal stress (oi) must not exceed the 
working stress (
< r y
) for the material.
ff! < (T y
Note: For bittle material, it gives satisfactory result. Yield criteria for 3D stress 
system,
a , = a v or a } = a[
Where, oy = Yield stress point in simple tension, and oy = Yield stress point in 
simple compression.
Stress&s on 
rectangular section
Maximum Principal Strain Theory (St. Venant's theory):
According to this theory, a ductile material begins to yield when the maximum 
principal strain at which yielding occurs in simple tension.
For 3D stress system,
ej,2,3< h r
a, 1 , N
h = —~ — (a\ + o -3 )
• m•
y
< ? 2 1 , .
ez= --------- (o - 3 + < T i)
s ms
cr. 1
e- = — ------(O ’ ! — 0 \J
s ms '
£
Yield point strain compressive
e f
~E
According to theory, ei = ey 
Yield criteria:
L ( ^ : + ^ ) = y
• m s E
And
For 2D system,
<Zl
E mE ~ E
(?i----- < a v
m "
Note: This theory can estimate the elastic strength of ductile material.
Maximum Shear Stress Theory (Guest &Tresca’s theory): According to this theory, 
failure of specimen subjected to any combination of loads when the maximum 
shearing stress at any point reaches the failure value equal to that developed at the 
yielding in an axial tensile or compressive test of the same material.
For 3D system:
Yielding criteria,
_ 1 , V _ 9 ,
• aa — _ t1 7 ! ai ' — -
In case of 2D: Oi - 03 = oy 
Yielding criteria, Oi - 02 = oy