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EN0175-22

# EN0175-22 - EN0175 11 21 06 Principle of minimum potential...

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EN0175 11 / 21/ 06 Principle of minimum potential energy (continued) The potential energy of a system is = S i i V i i V S u t V u f V w V d d d Principle of minimum potential energy states that for all kinematically admissible , the actual displacement field minimizes . i u V Example 1: g ρ x L We have shown in the beginning of the semester that the exact solution is: ( ) x L x E g u = 2 ρ . Now we discuss how to solve the same problem by using the principle of minimum potential energy. Procedure: 1) Pick any displacement such that ( ) ( ) 0 0 = = L u u . 2) Minimize for the chosen parameters. V An obvious choice is ( ) ( ) ( ) x f x L x x u = since this satisfies the clamped displacement boundary conditions for any . We can assume ( ) x f ( ) x f to be a polynomial function. ( ) N N x C x C x C C x f + + + + = L 2 2 1 0 To minimize the potential energy , we take ( ) N C C C C V , , , , 2 1 0 L 0 0 = C V , 0 1 = C V , …, 0 = N C V 1

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