EN0175 11 / 21/ 06 Principle of minimum potential energy (continued)The potential energy of a system is ∫∫∫−−=SiiViiVSutVufVwVdddPrinciple of minimum potential energy states that for all kinematically admissible , the actual displacement field minimizes . iuVExample 1: gρxLWe have shown in the beginning of the semester that the exact solution is: ()xLxEgu−=2ρ. Now we discuss how to solve the same problem by using the principle of minimum potential energy. Procedure: 1)Pick any displacement such that ( )( )00==Luu. 2)Minimize for the chosen parameters. VAn obvious choice is ( )()( )xfxLxxu−=since this satisfies the clamped displacement boundary conditions for any . We can assume ( )xf( )xfto be a polynomial function. ( )NNxCxCxCCxf++++=L2210To minimize the potential energy , we take ()NCCCCV,,,,210L00=∂∂CV, 01=∂∂CV, …, 0=∂∂NCV1
has intentionally blurred sections.
Sign up to view the full version.