Energy_06_Minimum_Potential_Energy

Energy_06_Minimum_Potential_Energy - Section 5.6 5.6 The...

This preview shows pages 1–3. Sign up to view the full content.

Section 5.6 Solid Mechanics Part I Kelly 208 5.6 The Principle of Minimum Potential Energy The principle of minimum potential energy follows directly from the principle of virtual work (for elastic materials). 5.6.1 The Principle of Minimum Potential Energy Consider again the example given in the last section; in particular re-write Eqn. 5.5.15 as 0 2 2 2 2 2 2 1 1 1 = + B B u L A E L A E Pu δ (5.6.1) The quantity inside the curly brackets is defined to be the total potential energy of the system, Π , and the equation states that the variation of Π is zero – that this quantity does not vary when a virtual displacement is imposed: 0 = Π (5.6.2) The total potential energy as a function of displacement u is sketched in Fig. 5.6.1. With reference to the figure, Eqn. 5.6.2 can be interpreted as follows: the total potential energy attains a stationary value (maximum or minimum) at the actual displacement ( 1 u ); for example, 0 Π for an incorrect displacement 2 u . Thus the solution for displacement can be obtained by finding a stationary value of the total potential energy. Indeed, it can be seen that the quantity inside the curly brackets in Fig. 5.6.1 attains a minimum for the solution already derived, Eqn. 5.5.17. Figure 5.6.1: the total potential energy of a system To generalise, define the “potential energy” of the applied loads to be ext W V = so that V U + = Π (5.6.3) The external loads must be conservative, precluding for example any sliding frictional loading. Taking the total potential energy to be a function of displacement u , one has 0 ) ( = Π = Π u du u d (5.6.4) ) ( u Π 1 u 1 u 2 u 2 u

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Section 5.6 Solid Mechanics Part I Kelly 209 Thus of all possible displacements u satisfying the loading and boundary conditions, the actual displacement is that which gives rise to a stationary point 0 / = Π du d and the problem reduces to finding a stationary value of the total potential energy V U + = Π .
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 01/20/2012 for the course ENGINEERIN 1 taught by Professor Staff during the Fall '11 term at Auckland.

Page1 / 5

Energy_06_Minimum_Potential_Energy - Section 5.6 5.6 The...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online