MEM427_Lecture_6_Weighted_Residual_Methods

MEM427_Lecture_6_Weighted_Residual_Methods -...

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MEM427 Introduction to Finite Element Method cture 6 Lecture 6 Weighted Residual Methods uckling (Stability) Analysis Buckling (Stability) Analysis Non Linear Analysis Chapter 6 Weighted Residual Method; Instability Analysis 1 MEM427 Introduction to Finite Element Method Recall simple beam problems p dx v d EI n al Equatio Differenti 4 4 (D.E.) the Solving 2 L 2 L L dx pv dx v d EI l Equation Variationa 0 2 2 2 2 ) ( Energy Potential Total the Minimizing 1 v v ex v D.E. can be expressed in a general form: u L ʺ V.E. can be expressed in a general form: i n v f Du   dx v v x F 0 , , operator al Differenti : 4 4 dx d EI D variable Dependent : v h Pi i l f Mi i Ttl P t ti l E tt t  function (forcing) Driving : variable Dependent : x p f x v u ʺ Functional ʺ : 2 2 pv v EI F Chapter 6 Weighted Residual Method; Instability Analysis 2 The Principle of Minimum Total Potential Energy states that the solution v ( x ) that satisfies the D.E. also minimizes the V.E. MEM427 Introduction to Finite Element Method Calculus of Variation L dx v v x F f Du 0 ʺ , , : V.E. ; : D.E. EI d 2 4 eams mple or From Calculus of Variation one can show that minimization of V.E. yields the so called Euler Lagrange equation (i.e., the D.E.)   pv v F p f v u dx D 4 2 , , , : beams simple For 0 2 2 v F dx d v F dx d v F pv v EI F 2 2 : Beams 0 4 4 p dx v d EI On the other hand, if the D.E. satisfies certain conditions, then it is possible to obtain the corresponding V.E. L L fd 1    ufdx dx u Du 0 0 2 p f v u x d D , , : Beams 4 4 L L vpdx dx v EI 0 0 2 Chapter 6 Weighted Residual Method; Instability Analysis 3 dx 2 (MEM660 and MEM661 Theory of Elasticity) MEM427 Introduction to Finite Element Method Approximate Solution Methods Based on Variational Principles L pp v d I 2 2 app app app dx x p x v dx EI 0 2 ) ( ) ( 2 Examples: A simply supported beam Recall beam problems L x A v v app sin 1 1 EI PL A A app 4 3 1 1 2 0 P x p v L x A L x A v v app 3 sin sin 3 1 3 PL PL a p app 3 3 2 2 v app EI A EI A A A pp pp 4 3 4 1 3 1 81 ; 0 ; 0 , 1 x L x B v app Chapter 6 Weighted Residual Method; Instability Analysis 4 This is also known as the “Ritz Method”

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MEM427 Introduction to Finite Element Method Variational Principles Other Examples: Thin plates h 4 4 4 3 p ( x , y )  y x p y w y x w x w v Eh , 2 1 12 : D.E 4 2 2 4 2 
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This note was uploaded on 11/14/2010 for the course MEM 427 taught by Professor Tein-mintan during the Fall '10 term at Drexel.

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MEM427_Lecture_6_Weighted_Residual_Methods -...

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