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MEM427_Lecture_6_Weighted_Residual_Methods

# MEM427_Lecture_6_Weighted_Residual_Methods - L2 L2...

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MEM427 Introduction to Finite Element Method Lecture 6 Weighted Residual Methods 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 h Mi i i i 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 i v n v ex v D.E. can be expressed in a general form: f Du L dx v v x F ʺ V.E. can be expressed in a general form: 0 , , operator al Differenti : 4 4 dx d EI D variable Dependent : EI v Th P i i l f Mi i T t l P t ti l E t t th t     function (forcing) Driving : variable Dependent : x p f x v u ʺ Functional ʺ : 2 2 pv v 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. pv v EI F p f v u d D 2 4 : beams simple For From Calculus of Variation one can show that minimization of V.E. yields the so called Euler Lagrange equation (i.e., the D.E.) dx 4 2 , , , 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 d D 1 ufdx dx u Du 0 0 2 p f v u dx d D , , : Beams 4 4 L L vpdx dx v EI 0 0 2 2 Chapter 6 Weighted Residual Method; Instability Analysis 3 (MEM660 and MEM661 Theory of Elasticity) MEM427 Introduction to Finite Element Method Approximate Solution Methods Based on Variational Principles L app d v d EI 2 2 ) ( ) ( ll b bl app app dx x p x v dx 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 A PL A app app 3 3 2 2 0 0 v app EI EI A A 4 3 4 1 3 1 81 ; ; , 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 h l h l Other Examples: Thin plates w w w Eh 4 4 4 3 p ( x , y ) y x p y y x x v , 2 1 12 : D.E 4 2 2 4 2    
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MEM427_Lecture_6_Weighted_Residual_Methods - L2 L2...

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