Energy method
- 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
- 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