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# ex15_040108 - EP 471 Engineering Problem Solving II...

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4/1/08 EP 471 -- Engineering Problem Solving II Exercise 15 Gaussian Quadrature and Work Equivalence in Finite Elements In EMA 405, we typically look at a few idealized problems where we convert a continuous load distribution into an equivalent discrete load set. Focusing now on eight- node quads, we can perform integrations in 2D problems that convert body forces to discrete loads or in 3D problems that convert surface tractions to discrete loads (8-node quads are the surfaces of 20-node hexahedral elements). One of our results is that for square elements, a constant pressure exerted on an 8-node quad produces a work- equivalent load set equal to -1/12 × pA on the corner nodes and +1/3 × pA on the midside nodes, where F = pA is the total resultant force on the face of the element (pressure × surface area). The form of the conversion produces force on node i equal to: η ξ η ξ η ξ d d J f p F i i det ) , ( ) , ( 1 1 1 1 ∫ ∫ + + = Here p is the distributed load on the surface (constant for uniform pressure), f i
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