28 Stress Recovery IFEM Ch 28 Slide 1 Introduction to FEM - PDF document

Introduction to FEM 28 Stress Recovery IFEM Ch 28 – Slide 1

Introduction to FEM Stress Recovery Processing phase has solved for node displacements from the (modified) master stiffness equations K u = f Postprocessing phase now starts to get derived quantities. Among them are internal forces and stresses. The process of computing stresses from node displacements is called stress recovery . IFEM Ch 28 – Slide 2

Introduction to FEM General Comments Stresses recovered from low order elements (e.g. 3-node triangles and 4-node quads) often display large interelement jumps . In-plane bending situations are particularly troublesome Jumps can be eliminated by interelement averaging at nodes This usually improves the stress quality at interior nodes, but may not be effective at boundary nodes. Stress recovery over quadrilateral elements can be improved by extrapolation from Gauss sample points � IFEM Ch 28 – Slide 3

Introduction to FEM The Berkeley Cantilever IFEM Ch 28 – Slide 4

Introduction to FEM Nodal Stress Averaging IFEM Ch 28 – Slide 5

Introduction to FEM Gauss Elements η (b) (a) 3 3 3' η 4 ' 4 4' (e') ξ (e) ξ ' 1' 2' 1 1 2 2 IFEM Ch 28 – Slide 6

Introduction to FEM Table 28.1 Natural Coordinates of Bilinear Quadrilateral Nodes ξ ′ η ′ ξ ′ η ′ Corner ξ η Gauss ξ η node node √ √ √ √ 1 − 1 − 1 3 3 1’ − 1 / 3 − 1 / 3 − 1 − 1 − − √ √ √ √ 2 + 1 − 1 3 3 2’ + 1 / 3 − 1 / 3 + 1 − 1 + − √ √ √ √ 3 + 1 + 1 3 3 3’ + 1 / 3 + 1 / 3 + 1 + 1 + + √ √ √ √ 4 − 1 + 1 3 3 4’ − 1 / 3 + 1 / 3 − 1 + 1 − + Gauss nodes, and coordinates ξ ′ and η ′ are defined in §28.4 and Fig. 28.1 IFEM Ch 28 – Slide 7

Introduction to FEM Extrapolation to the Corner Points Shape functions of "Gauss element" N ( e ′ ) = 1 4 ( 1 − ξ ′ )( 1 − η ′ ), 1 N ( e ′ ) = 1 4 ( 1 + ξ ′ )( 1 − η ′ ), 2 N ( e ′ ) = 1 4 ( 1 + ξ ′ )( 1 + η ′ ), 3 N ( e ′ ) = 1 4 ( 1 − ξ ′ )( 1 + η ′ ). 4 To extrapolate, replace the ξ ' and η ' corner coordinates of the actual element: √ √ 1 + 1 − 1 1 − 1 − 1 3 3   w ′ w 1     2 2 2 2 √ √ 1 − 1 1 + 1 − 1 1 − 1 3 3 w ′ w 2   2 2 2 2 2 √ √    =     1 − 1 − 1 1 + 1 − 1 w ′ w 3  3 3     3 2 2 2 2 √ √ w ′ w 4 − 1 1 − 1 − 1 1 + 1 3 3 4 2 2 2 2 IFEM Ch 28 – Slide 8

Introduction to FEM Other "Gauss Element" Configurations (a) (b) 3 7 (c) 3 3 7 3' 7' 3' 4 3' 5 7' 4' 4 4' 6' 6 9 6 8' 6 6' 9 9' 8' 8 8 2' 2' 5' 2 1' 1' 2' 5' 1' 2 1 5 1 2 5 1 4 IFEM Ch 28 – Slide 9