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MEM427_Lecture_3_Energy_Principles

MEM427_Lecture_3_Energy_Principles - OutlineofLecture...

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MEM427 Introduction to Finite Element Method Lecture 3 Energy Principles Chapter 3 Energy Principles 1 MEM427 Introduction to Finite Element Method Outline of Lecture •Strain Energy and Strain Energy Density •Castigliano’s Theorems •Principle of Minimum Total Potential Energy Chapter 3 Energy Principles 2 MEM427 Introduction to Finite Element Method Strain Energy For Axially Loaded Members W : Work done by the load 0 1 1 d P W 1 1 d P W U U : Strain energy stored in the member If no energy is dissipated, then Chapter 3 Energy Principles 3 0 MEM427 Introduction to Finite Element Method Strain Energy For Axially Loaded Members For Linearly Elastic Materials P P 2 P W R ll f i ll l d d b P Inelastic L EA P EA PL or , Recall for axially loaded members Elastic strain energy   EA L P P W U 2 2 2 EA 2 strain energy Chapter 3 Energy Principles 4 L P W U 2 2 Elastic deformation Permanent deformation
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MEM427 Introduction to Finite Element Method Example: Determine the displacement at joint B cos H L L EA EA L P U 2 2 2 2 2 B P W L Work Done by P: B AB AB AB AB AB AB A E L S Strain Energy in i th Member: Free body di BC S BA S i i i i i A E L S U 2 2 EA L S U U i i 2 2 2 2 1 diagram for joint B P B 0 sin sin S S F Total Strain Energy: 2 H P 0 cos cos P S S F BC BA y PH P BC BA x 3 cos 4 EA Chapter 3 Energy Principles 5 U W 3 cos 2 EA B cos 2 S S S BC BA MEM427 Introduction to Finite Element Method Example HW #1a Revisit B C D ft .5 1 ft .0 1 j P i P P A 45 sin 45 cos : at Force j v i u d A A A A : at nt Displaceme ft 2 2 6 in 0 8 psi 10 9 . 1 A E A i i d P P W 2 1 2 1 : Done Work 45 sin 45 cos 2 1 A A v P u P A x, u y, v 45 . 3 2 2 : Energy Strain i i E A L S U lb in 1606 . 0 in 83 . 26 in, 0 . 24 in, 0 . 30 AD AC AB L L L Member S (lb) L (in) U (in lb) AB 364.30 30.00 0.1310 1 i i i P = 400 lb ll h l i f in 10 1894 0 in 10 9459 . 0 3 3 A v u lb 0 . 120 lb 3 . 364 AC AB S S AC 120.00 24.00 0.0114 AD 143.70 26.83 0.0182 Recall the solution for HW #1a: Chapter 3 Energy Principles 6 . A lb 7 . 143 AD S W U lb in 0.1606 i U MEM427 Introduction to Finite Element Method Homework Assignment #3(a) Use the solution of Lab #1 to verify that the strain energy ( U ) stored in the truss is equal to the work done ( W ) by the loads.
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