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Unformatted text preview: MAE M168/CEE M135C Introduction to Finite Element Methods Lecture 2 Spring Models Review: Problemsolving strategy with FEM 1. Define a model • Mathematical idealization of the real physical problem • Choose what details to include/omit: Static vs. dynamic? Beam or 3 D solid? Loads/boundary conditions? Connections/contact? Elastic vs. elastoplastic? Composite vs. homogeneous? . reliminary Analysis 2. Preliminary Analysis • Simplified “handcalculation” version of the model 3. FEA solution 4. Check answers • Compare to prelim. analysis/experiment, think and evaluate 5. Revise/refine Only step 3 is done by the software! Rest is up to you! Review: Basic FEA procedure 1. Discretize continuous body into elements 2. Approximate local behavior of elements in terms of discrete DOFs 3. Patch together, or assemble elements to form a global system of algebraic equations and solve for discrete DOFs. Spring Models h Why are spring models useful? Simple idealization of a structure. Example: 1D structure P Al Steel k (1) k (2) D 1 D 2 P Spring model: Basic FEA procedure 1. Discretize continuous body into elements 2. Approximate local behavior of elements in terms of discrete DOFs 3. Patch together, or assemble elements to form a global system of algebraic equations and solve for discrete DOFs. P Al Steel Discretize Body Q i = Global Nodal applied loads D i = Global Nodal displacements Q 2 k (1) k (2) D 1 D 2 Q 1 Q o D o node 1 node 3 node 2 element 1 element 2 Basic FEA procedure 1. Discretize continuous body into elements 2. Approximate local behavior of elements in terms of discrete DOFs 3. Patch together, or assemble elements to form a global system of algebraic equations and solve for discrete DOFs. Local Element Behavior i j ( ) e k ( ) e i f ( ) e j f ( ) e d ( ) e d Local equilibrium: ( ) ( ) e e i j f f + = i j ( ) ( ) e e j i f f f ⇒ =  = Hooke’s law: k l f Δ = Compatibility of displacements: ( ) ( ) e e j i l d d Δ = ( ) ( ) ( ) ( ) e e e j i j i k d d f f ∴ = =  ( ) 2 equations Local Element Behavior Or ( ) ( ) 1 e e f ( ) ( ) ( ) ( ) e e e j i j i k d d f f ∴ = =  ( ) 2 equations ( ) ( ) ( ) 1 1 1 1 i i e e e j j d f k d f  =  (e) k (e) d (e) f (e) : element (local) stifness matrix k (e) : element (local) displacement vector d (e) : element (local) force vector f Basic FEA procedure 1. Discretize continuous body into elements 2. Approximate local behavior of elements in terms of discrete DOFs 3. Patch together, or assemble elements to form a global system of algebraic equations and solve for discrete DOFs. Assembly (1) (1) (1) (1) (1) 1 1 1 1 i i j j d f k d f  =  1 e = (2) (2) (2) (2) (2) 1 1 1 1 i i j j d f k d f  =  2 e = Global Nodes (1) (1) (1) 1 1 1 i o f D k  =...
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This note was uploaded on 08/09/2011 for the course MAE 168 taught by Professor Klug during the Spring '11 term at UCLA.
 Spring '11
 Klug

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