Lecture #6
Energy concepts.
Calculation of displacements
TOPIC OF THE LECTURE
Energy concepts:
•Virtual work concept
•Strain energy
•Castigliano’s theorem
•Mohr’s formula
Specific calculations of displacements:
•Trusses
•Actuated structures (temperature, piezoelectric and
shape memory alloy displacements)
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PRINCIPLE OF VIRTUAL WORK
The principle claims that for the state of equilibrium
the sum of works of all forces on any possible (virtual)
displacement is zero. This principle is widely used
when a problem is solved using variational calculus.
 Wtotal  0
For linear displacement
W  F   r
For rotational displacement
 W  M  
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STRAIN ENERGY AND COMPLEMENTARY ENERGY
dU
P
dy
dC
y
dP
2
Q y2dx
Qz2 dx M x2dx M y dx M z2dx
N x2dx
dU 
 Ky 
 Kz 



2 EA
2GA
2GA 2GI x
2 EI y
2 EI z
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CASTIGLIANO’S THEOREM
Theorem states that the displacement due to a certain
force is equal to the derivative of complementary
energy by this force. For elastic structures, however,
strain energy could be used instead of complementary
energy:
dU

dFd
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CASTIGLIANO’S THEOREM
The partial case for elastic trusses is:
dSi
Si 
 Li
dFd
Si  S i  Li
dU



dFd
Ei  Ai
E i  Ai
i
i
Where S i could be considered as a force in the
member due to applied unity force corresponding to
the displacement .
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MOHR’S INTEGRAL
In contrast to Castigliano’s theorem which is
applicable for general problem, the Mohr’s integral is
applicable for elastic problems only.
Qy  Qy
Qz  Qz
Nx  Nx

dx  K y  
dx  K z  
dx 
EA
GA
GA
My  My
Mz  Mz
Mx  Mx

dx  
dx  
dx
GI x
EI y
EI z
Displacements from shear forces are usually
negligibly small comparing to those from bending
moments.
Displacements from normal forces are smaller than
from bending moments, they are used for trusses.
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DISPLACEMENTS IN SHAPE MEMORY ALLOYS
Shape memory alloys (SMA) demonstrate quite
complex behavior.
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DISPLACEMENTS IN PIEZOELECTRICS
The total strain is equal to the sum of mechanical and
actuated strain:
 total   mechanical   piezoelectric
The actuated strain is directly proportional to electric
field E:
 piezoelectric  d  E
Typically, value of piezoelectric strain is
 piezoelectric  0.2  10
3
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THERMAL ACTUATION OF DISPLACEMENTS
The total strain is equal to the sum of mechanical and
actuated strain:
 total   mechanical   thermal
The actuated strain is directly proportional to change
in temperature T:
 thermal  a  T
The thermal strain highly depends on change in
temperature and the type of material (a = 5·10-6 for
wood, a = 10..25·10-6 for most metals, a = 200·10-6
for plastics).
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THERMAL ACTUATION OF DISPLACEMENTS
The case when beam is subjected to the temperature
linearly distributed along the height:
  a 
t upper  t lower
2
 N  dx   a 
tupper  t lower
2
 M  dx
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WHERE TO FIND MORE INFORMATION?
For energy-related concepts, I recommend
T.H.G. Megson. An Introduction to Aircraft Structural Analysis. 2010
Chapters 4 and 5
For Mohr’s integral, refer to your Mechanics of Materials
course.
… Internet is boundless …
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TOPIC OF THE NEXT LECTURE
Statically indeterminate structures. Method of forces
All materials of our course are available
at department website k102.khai.edu
1. Go to the page “Библиотека”
2. Press “Structural Mechanics (lecturer Vakulenko S.V.)”
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