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

SLIDE 1

Introduction to FEM

28

Stress Recovery

IFEM Ch 28 – Slide 1

SLIDE 2

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.

Introduction to FEM

IFEM Ch 28 – Slide 2

SLIDE 3

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

• Introduction to FEM

IFEM Ch 28 – Slide 3

SLIDE 4

The Berkeley Cantilever

Introduction to FEM

IFEM Ch 28 – Slide 4

SLIDE 5

Nodal Stress Averaging

Introduction to FEM

IFEM Ch 28 – Slide 5

SLIDE 6

Gauss Elements

(a) (b) 1 1' 1 2 2' 2 3 3' 3 4 4' 4 ξ η ξ η (e) ' ' (e')

Introduction to FEM

IFEM Ch 28 – Slide 6

SLIDE 7

Table 28.1 Natural Coordinates of Bilinear Quadrilateral Nodes

Corner ξ η ξ ′ η′ Gauss ξ η 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

Introduction to FEM

IFEM Ch 28 – Slide 7

SLIDE 8

N (e′)

1

= 1

4(1 − ξ ′)(1 − η′),

N (e′)

2

= 1

4(1 + ξ ′)(1 − η′),

N (e′)

3

= 1

4(1 + ξ ′)(1 + η′),

N (e′)

4

= 1

4(1 − ξ ′)(1 + η′).

   w1 w2 w3 w4    =     1 + 1

2

√ 3 − 1

2

1 − 1

2

√ 3 − 1

2

− 1

2

1 + 1

2

√ 3 − 1

2

1 − 1

2

√ 3 1 − 1

2

√ 3 − 1

2

1 + 1

2

√ 3 − 1

2

− 1

2

1 − 1

2

√ 3 − 1

2

1 + 1

2

√ 3        w′

1

w′

2

w′

3

w′

4

  

Extrapolation to the Corner Points

Shape functions of "Gauss element" To extrapolate, replace the ξ' and η' corner coordinates of the actual element:

Introduction to FEM

IFEM Ch 28 – Slide 8

SLIDE 9

Other "Gauss Element" Configurations

1 1 2 2 3 3 4 4 5 5 6 6 7 8 9 1 2 3 4 5 6 7 8 9 1' 2' 3' 4' 5' 6' 7' 8' 1' 2' 3' 1' 2' 3' 4' 5' 6' 7' 8' 9' (a) (b) (c)

Introduction to FEM

IFEM Ch 28 – Slide 9