17_Energy Methods

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

This preview shows pages 1–4. Sign up to view the full content.

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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern