Energy methods applications

1. 15 November 2016
Energy Methods Applications
CE 527
THEORY OFElasticity
Mohanad Ibrahim Al-Samaraie
S.N.: 201568584
College of Engineering
Assist. Prof. Dr. Nildem TAYŞİ

2. Introduction
• Energy methods are used widely to obtain solutions to elasticity
problems and determine deflections of structures.
• The deflection of joints on a truss or points on a beam or shaft can be
determined using energy methods. we will first define the work
caused by an external force and couple moment and show how to
express this work in terms of a body’s strain energy.
• In mechanics, a force does work when it undergoes a displacement
dx that is in the same direction as the force.

3. External Force and Strain Energy
• A uniform rod is subjected to a slowly increasing
load
workelementarydPdUe =∆=
energystrainworktotaldPUe ==∆= ∫
∆
0
∆=∆=∆∆= ∫
∆
12
12
12
1
0
PkdkUe
• In the case of a linear elastic deformation,

4. ∆′′+∆′+∆= PPPUe 2
1
2
1
The work done by P´ is equal to the gray shaded triangular area
and the work done by P represents the dark-blue shaded
rectangular area

5. 112
1
0
1
θθ
θ
MdMUe ==∫
• For a shaft subjected to a torsional load,
112
1
0
1
φφ
φ
TdTUe ==∫

6. Internal Force and Strain Energy Density

7. • When a body is subjected to shear stress,

8. I
yM
x =σ
• For a beam subjected to a bending load,
∫∫ == dV
EI
yM
dV
E
U x
i 2
222
22
σ
• Setting dV = dA dx,
dx
EI
M
dxdAy
EI
M
dxdA
EI
yM
U
L
L
A
L
A
i
∫
∫ ∫∫ ∫
=








==
0
2
0
2
2
2
0
2
22
2
22
• For an end-loaded cantilever beam,
EI
LP
dx
EI
xP
U
PxM
L
i
62
32
0
22
==
−=
∫

9. J
T
xy
ρ
τ =
∫∫ == dV
GJ
T
dV
G
U
xy
i 2
222
22
ρτ
• For a shaft subjected to a torsional load,
• Setting dV = dA dx,
∫
∫ ∫∫∫
=








==
L
L
A
L
A
i
dx
GJ
T
dxdA
GJ
T
dxdA
GJ
T
U
0
2
0
2
2
2
0
2
22
2
22
ρ
ρ
• In the case of a uniform shaft,
GJ
LT
Ui
2
2
=

10. • The strain energy stored in members subjected to
several types of loads, the normal and shear
stress components, can be obtained from
• The total strain energy in the body is therefore
• The strains can be eliminated by using the generalized
form of Hooke’s law given by

11. • After substituting and combining terms, we have

12. • For adiabatic conditions (no heat flow) and static equilibrium
(kinetic energy=0), the variation in work of the external force
is equal to the variation of internal energy
The work performed on a mechanical system by external forces plus
the heat that flaws into the system from the outside equals the
increase in internal energy plus the increase in kinetic energy
Energy Methods in Elasticity

13. • From above two equation, we obtain

15. Elasticity and Complementary Energy Density

16. Castigliano’s Theorem
• This method, which is referred to as
Castigliano’s theorem, applies only to bodies
that have constant temperature and material with
linear-elastic behaviour
),.......,,( 21 nei PPPfUU ==
• If any one of the external forces, say Pj , is
increased by a differential amount dPj , the
internal work will also be increased, such that
the strain energy becomes

17. and
j
j
j
j
j
j
T
U
M
U
P
U
x
∂
∂
=
∂
∂
=
∂
∂
= φθ

18. Deflections By Castigliano’s Second Theorem
• In the case of a beam,
∫
∫∫








∂
∂
=∆
∂
∂
=∆=
L
j
j
L
j
j
L
EI
dx
P
M
M
dx
EI
M
P
dx
EI
M
U
0
0
2
0
2
22
• For a truss,
EA
L
P
F
F
EA
LF
PEA
LF
U
i
i
j
i
n
i
j
n
i i
ii
j
j
n
i i
ii
i








∂
∂
=∆
∂
∂
=∆=
∑
∑∑
=
==
1
1
2
1
2
22

19. Principle of Virtual Work
• We will use it to obtain the displacement and slope at a
point on a deformable body
• The principle of virtual work was developed by John
Bernoulli in 1717,
• Consider the body to be of arbitrary shape and to be
subjected to the “real loads” P1, P2, and P3
• It is assumed that these loads cause no movement of the
supports
• To determine the displacement Δ of point A on the body,
place an imaginary or “virtual” force P´ on the body at
point A, such that P´ acts in the same direction as Δ
• P´ to have a “unit” magnitude; that is, P´ = 1 and create an
internal virtual load u in a representative element or fiber
of the body

20. • The external virtual work is then equal to the internal virtual
work done on all the elements of the body.

21. • We can express strain energy density as

22. 1. A.P. Boresi, R.J. Schmidt, “Advanced Mechanics of Materials”,6th
edition, J
Wiley 2003
2. R.C Hibbeler, “Mechanics of Materials”, 9th
edition, Pareson 2014.
3. Ferdinand P. Beer E., Russell Johnston, Jr., John T. DeWolf, David F. Mazurek,
“ Mechanics of Materials”,7th
edition, McGraw-Hill Education,2015.
4. J. M. Gere, S. P. Timoshinko, “Mechanics of Materials”, 4th
edition, PSW
Publishing 1997.
5. Teodor M. Atanackovic, Ardeshir Guran, “Theory of Elasticity for Scientists
and Engineers”, Springer 2000.
6. Lecture Note(Energy methods)
References