Chapter 9_FEM_NEW - CHAPTER 9 FINITE ELEMENT MODELING The...

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1 CHAPTER 9 FINITE ELEMENT MODELING The finite element method is a very powerful tool for modeling and analysis of various structures (aircraft, spacecraft, launch vehicles, engines, automobiles, bridges, buildings, machine components) with complicated geometry. The finite element method transforms a structure into a system with a finite number of degrees of freedom (DOFs). In this chapter, we will consider finite element modeling and analysis of slender structures under uniaxial load, torque or bending loads. This will help us understand the fundamentals of the finite element analysis, while keeping the number of degrees of freedom small.
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2 Single DOF system kq F = Multi DOF systems For complicated structures, such as aircraft or spacecraft, N N N 1 1 × × × = N N N N F q K Examples: Boeing 777 - 255, 000 DOF Lockheed Martin F-22 fighter aircraft - 140,000 DOF Sikorsky/Boeing Comanche Helicopter - 149,000 DOF P & W F-119 jet engine - 24,000 DOF Space Station Freedom - 1000,000 DOF A solid body with infinite number of degrees of freedom Equilibrium eq. involving a finite number of degrees of freedom (DOF) N : the number of degrees of freedom in the model Finite element modeling ⎯→ ⎯⎯⎯⎯ k F, q
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3 Equilibrium via Energy Approach For structures of complicated geometry, a more systematic approach is needed to derive the finite element equilibrium equation. Various methods are available for this purpose. For example, the equilibrium equation of a structure can be obtained from the extremum condition of a scalar quantity, called total potential energy, defined as UW Π= − where Π : total potential energy U : strain energy W : external work potential Strain Energy For elastic solids, the work done by ‘external’ force is stored as an internal energy called ‘strain energy’ First law of thermodynamics (on energy conservation). (1) Linear spring under applied force Work done by the applied force F (as the displacement q increases from q = 0 to ± q ) is 2 ˆ 0 2 ˆ 0 ˆ 0 ˆ 2 1 2 1 q K Kq dq Kq Fdq q q q q q q = = = = = = (1) Strain energy Uq K q Fq ( ± ) ± ± ± == 1 2 1 2 2 (2) So, strain energy is equal to the shaded area. In general, dropping the ‘hat’ symbol for convenience, 2 1 () 2 Kq = ( 3 ) K F, q 1 K F q F Kq = F q ± q ± F
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4 The principle of stationary total potential energy Example 1 : Consider a SDOF system of a spring under applied force F of given magnitude (say 100 lbs ). Strain energy corresponding to displacement q, Uq Kq () = 1 2 2 ( 1 ) Introduce a scalar function of Wq () called 'external work potential' defined as W qF q = ( 2 ) Note that this is not the work done by the external force F . Introduce also a scalar function Π q called 'total potential energy' defined as follows: Π qU qW q K q =−= 1 2 2 (3) Then, for the displacement q that corresponds to the extremum, 0 UW U F qq q q U FK q F q ∂Π =→ = ∂∂ →= (4) That is, the stationary or extremum condition of Π q leads to the equilibrium equation.
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Chapter 9_FEM_NEW - CHAPTER 9 FINITE ELEMENT MODELING The...

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