4-Trusses-2010s-rev1

4-Trusses-2010s-rev1 - MAE M168/CEE M135C Introduction to...

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© William Klug MAE M168/CEE M135C Introduction to Finite Element Methods Lecture 4 Truss analysis

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© William Klug Summary & Review 3. Assemble global equilibrium equations (code numbers) 1. Discretize Body ( ) o q x q = 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) 2. Local element behavior Element Code Number (1) 01 (2) 12 (3) 23 (4) 34 ( ) ( ) ( ) ( ) ( ) (1) 1 1 1 (1) (1) 1 1 0 1 0 ... 1 E A k k k l k k = = k ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + = + + 1 2 2 2 2 3 3 3 3 4 4 4 4 1 2 3 4 1 2 3 4 k k k 0 0 k k k k 0 0 k k k k 0 0 k k K Q = Q 1 Q 2 Q 3 Q 4 1 2 3 4 = f qj 1 ( ) + f qi 2 ( ) f qj 2 ( ) + f qi 3 ( ) f qj 3 ( ) + f qi 4 ( ) f qj 4 ( ) = q 0 L 4 q 0 L 4 q 0 L 4 q 0 L 8 f qi f qj f i f j = k ( e ) k ( e ) k ( e ) k ( e ) d i d j f qi f qj f ( e ) = k ( e ) d ( e ) f q ( e )
© William Klug Trusses Stress: P A σ = Hooke's Law: (1-D) E σ = ε Discretize a Truss ( ) j i P E d d A L = Strain: j i d d L L L Δ ε = = P P d i d j L ( ) j i EA P d d L = i P f = j f P = 1 1 1 1 i i j j d f k d f = Local Element behavior 4 D 8 D 1 D 2 D 3 D 5 D 6 D 7 D ( ) 2 ( ) 1 ( ) 3 ( ) 4 ( ) 6 ( ) 5 ( ) 7

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© William Klug Truss Analysis Technically not a FEA problem because no approximation is made, solution is exact. Truss: bars connected at freely-rotating joints. Use simple 1-D bar stiffness relation. 1 1 1 1 i i j j d f k d f = = EA k l The catch: bars are assembled in different orientations need to use different coordinate systems.