3b-directFEA-2011s (1)

# 3b-directFEA-2011s (1) - MAE M168/CEE M135C Introduction to...

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MAE M168/CEE M135C Introduction to Finite Element Methods Lecture 3b FEA in 1-D: direct formulation

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k (1) k (2) D 1 D 2 Q 1 Q o Q 2 D o P Al Steel 1. Discretize Body 2. Local element behavior ( ) e ( ) ( ) ( ) ( ) ( ) 1 1 1 1 e e i i e e e j j d f k d f - = - i j k ( ) e i f ( ) e j f ( ) e i d ( ) e j d 3. Assemble global equilibrium equations (code numbers) (1) (1) (1) (1) (1) k - = - k k k k 0 1 0 1 (2) (2) (2) (2) (2) k - = - k k k k 1 2 1 2 Element Code 1 0 1 2 1 2 k (1) + k (2) - k (2) - k (2) k (2) D 1 D 2 = Q 1 Q 2
What about continuously varying and distributed properties? k (1) k (2) D 1 D 2 Q 1 Q o Q 2 D o P Al Steel “Naturally Discrete” Body ( ) o q x q = ( ) 2 is constant o x E x E L A = - “Naturally Continuous” Body ?

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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.
Discretize Body (0) 0 = u ( ) o q x q = ( ) 2 is constant o x E x E L A = - R 0 = o D 1 D 2 D 3 D 4 D 1 k 2 k 3 k 4 k o Q 1 Q 2 Q 3 Q 4 Q (1) (2) (3) (4) Element Code Number (1) 01 (2) 12 (3) 23 (4) 34 How do we define ? k i 's How do we define ? 's i Q

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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.
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3b-directFEA-2011s (1) - MAE M168/CEE M135C Introduction to...

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