Problem 6.8-2If element thickness can vary and is computed as t= ΣNitifrom nodal values ti, what order Gaussquadrature is needed to compute the exact volume of (a) a four-node plane element, and (b) aneight-node plane element?Solution:Volume1−1η1−1ξJt⌠⎮⌡d⌠⎮⌡d=t1niNiti⋅()∑==JΣNiξ, xi⋅ΣNiη, xi⋅ΣNiξ, yi⋅ΣNiη, yi⋅⎛⎜⎝⎞⎟⎠=(a) For a 4-node element, each of the shape functions are 1st order in ξand η.Thus the determinate of [J] is a polynomial with ξand ηat most 1st power.Each of the Niare at most 1st order in ξand ηThusJt⋅fξ2η2, =The order of polynomial integrated exactly is:
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This note was uploaded on 01/11/2011 for the course MAE 5020 taught by Professor Folkman during the Fall '10 term at Utah State University.