17_Energy Methods - 5.1 GENERAL COMMENTS ON EQUILIBRIUM AND...

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2 5.1 GENERAL COMMENTS ON EQUILIBRIUM AND ENERGY METHODS TWO WAYS OF APPROACHING PROBLEMS IN STRUCTURAL MECHANICS. BOTH GIVE SAME RESULT, BUT CONCEPTUALLY COMPLETELY DIFFERENT. a) EQUILIBRIUM METHODS: ASSUME EVERY PART OF STRUCTURE IS IN EQUILIBRIUM (D'ALEMBERT'S FORCES INCLUDED FOR DYNAMIC PROBLEMS). FROM EQUILIBRIUM, STRESS-STRAIN, AND STRAIN-DISPLACEMENT EQUATIONS, FIND SOLUTION TO BOUNDARY VALUE PROBLEM, i.e. FIND DISPLACEMENTS ) t , ( x u (AND HENCE ALSO STRESSES). b) ENERGY METHODS: (ROUGHLY SPEAKING) OF ALL POSSIBLE DEFORMED SHAPES OF A BODY PERMITTED BY THE CONSTRAINTS, DETERMINE THE ACTUAL DEFORMED SHAPE ) t , ( x u BY SELECTING THE ONE WITH THE MINIMUM POSSIBLE TOTAL ENERGY. ADVANTAGES OF ENERGY METHODS: (i) SPECIFIC PROBLEMS EASIER (e.g. FINDING A PARTICULAR UNKNOWN FORCE OR DISPLACEMENT IN A COMPLICATED STRUCTURE). (ii) LOGICAL STARTING POINT FOR POWERFUL APPROXIMATION METHODS. (iii) CONSISTENT DERIVATION OF EQUILIBRIUM EQUATIONS AND BOUNDARY CONDITIONS. (iv) MINIMUM SMOOTHNESS OF CANDIDATE DISPLACEMENT FIELDS REQUIRED FOR APPROXIMATE METHODS.
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3 5.2 FINITE VS INFINITE DEGREE OF FREEDOM PROBLEMS = TOTAL POTENTIAL ENERGY U = (INTERNALLY STORED) ELASTIC ENERGY H = - (WORK DONE BY EXTERNAL LOADS) = U + H FINITE DEGREE OF FREEDOM SYSTEMS EXAMPLE 1 1 D.O.F. SYSTEM Px x K 2 1 x K 2 1 H U 2 2 2 1 + = + = MINIMIZE WITH RESPECT TO D.O.F. x. ( ) ) M EQUILIBRIU ( x K K P 0 P x K x K dx d 2 1 2 1 + = = + = x K K P
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4 EXAMPLE 2 2 D.O.F. SYSTEM i BAR OF ELONGATION i = i i i sin v cos u + = = U + H = E i A i 2 l i i ( ) 2 i P cos u P sin v MINIMIZE WITH RESPECT TO D.O.F. u, v.
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