Lecture #7Energy concepts.Calculation of displacements

TOPIC OF THE LECTURE

2

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)

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.

For linear displacement

For rotational displacement

3

PRINCIPLE OF VIRTUAL WORK

0totalW

W F r

W M

4

STRAIN ENERGY AND COMPLEMENTARY ENERGY

dUP

dy

dCy

dP

2 22 22 2

2 2 2 2 2 2y yz zx x

y zx y z

Q dx M dxQ dx M dxN dx M dxdU K K

EA GA GA GI EI EI

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:

5

CASTIGLIANO’S THEOREM

d

dU

dF

The partial case for elastic trusses is:

Where could be considered as a force in the member due to applied unity force corresponding to

the displacement .6

CASTIGLIANO’S THEOREM

ii i

id i i

i id i i i i

dSS L

dF S S LdU

dF E A E A

iS

In contrast to Castigliano’s theorem which is applicable for general problem, the Mohr’s integral is applicable for elastic problems only.

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. 7

MOHR’S INTEGRAL

y y z zx x

y z

y y z zx x

x y z

Q Q Q QN Ndx K dx K dx

EA GA GA

M M M MM Mdx dx dx

GI EI EI

8

Shape memory alloys (SMA) demonstrate quite complex behavior.

DISPLACEMENTS IN SHAPE MEMORY ALLOYS

The total strain is equal to the sum of mechanical and actuated strain:

The actuated strain is directly proportional to electric field E:

Typically, value of piezoelectric strain is

9

DISPLACEMENTS IN PIEZOELECTRICS

total mechanical piezoelectric

piezoelectric d E

30.2 10piezoelectric

The total strain is equal to the sum of mechanical and actuated strain:

The actuated strain is directly proportional to change in temperature T:

The thermal strain highly depends on change in temperature and the type of material (= 5·10-6 for wood, = 10..25·10-6 for most metals, = 200·10-6 for plastics).

10

THERMAL ACTUATION OF DISPLACEMENTS

total mechanical thermal

thermal T

The case when beam is subjected to the temperature linearly distributed along the height:

11

THERMAL ACTUATION OF DISPLACEMENTS

2 2upper lower upper lowert t t t

N dx M dx

WHERE TO FIND MORE INFORMATION?

12

For energy-related concepts, I recommendT.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 …

TOPIC OF THE NEXT LECTURE

13

Statically indeterminate structures. Method of forces

All materials of our course are availableat department website k102.khai.edu

1. Go to the page “Библиотека”2. Press “Structural Mechanics (lecturer Vakulenko S.V.)”