Lecture #7 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) 2

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. For linear displacement For rotational displacement 3

STRAIN ENERGY AND COMPLEMENTARY ENERGY 4

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: 5

CASTIGLIANO’S THEOREM 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 d. 6

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. 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

DISPLACEMENTS IN SHAPE MEMORY ALLOYS Shape memory alloys (SMA) demonstrate quite complex behavior. 8

DISPLACEMENTS IN PIEZOELECTRICS 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

THERMAL ACTUATION OF DISPLACEMENTS The total strain is equal to the sum of mechanical and actuated strain: The actuated strain is directly proportional to change in temperature DT: 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). 10

THERMAL ACTUATION OF DISPLACEMENTS The case when beam is subjected to the temperature linearly distributed along the height: 11

… Internet is boundless … 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 … 12

Statically indeterminate structures. Method of forces 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.)” 13