Co-rotational Formulation for Sandwich Plates and Shells

Yating Liang, Bassam A. Izzuddin

COMPUTATIONAL STRUCTURAL MECHANISM GROUP (CSM)

DEPARTMENTAL OF CIVIL AND ENVIRONMENTAL ENGINEERING

IMPERIAL COLLEGE LONDON

22ND ACME CONFERENCE ON COMPUTATIONAL MECHANICS - 2-4 APRIL - EXETER - UK

Background

Sandwich structures in civil engineering:

Insulation walls Roof panels Curtain wall glazing

Outline of Presentation

1 Element Formulation

1.1 Displacement fields

1.2 Transverse shear stress

1.3 Co-rotational framework

1.4 Shell coordinate system

1.5 Layer-wise theory

2 Numerical examples

Zigzag Displacement Fields

• Four orthogonal through-thickness displacement modes:

• Seven displacement parameters per node

• Five basic nodal freedoms

• Two additional freedomsi i i xi yi(u ,v ,w ,θ ,θ )

xi yi( , )

• Modulus ratio

Transverse Shear Stress Through Thickness

R=105 R=106R=102R=10

face

core

ER=

E

Bisector Co-rotational Framework

• Local x- and y- axes are set to be the bisectors of the two element diagonals in both the initial undeformed and the current deformed configuration.

• Basic freedoms are defined in this co-rotational system.

i i i xi yi(u ,v ,w ,θ ,θ )

Izzuddin & Li (2004)

Shell Coordinate System

• Additional freedoms are defined in local shell coordinate system.

• Two considerations:• Computational efficiency• Consideration of composite materials

xi yi( , )

α

x

y

Additional freedoms are defined in local shell coordinate system.

Shell Coordinate System

xi yi( , )

z xyn

Additional freedoms are defined in local shell coordinate system.

Shell Coordinate System

xi yi( , )

nz

x

y

α

Consideration of Composite Materials

α

β x-axis of element coordinate system

With the use of the shell coordinate system, the relative orientation of the composite material fiber relative to the element coordinate system is readily determined.

Layer-wise Theory

Hellinger-Reissner Variation Principle

w σx σy τxz τyz τxy0

0.2

0.4

0.6

0.8

1

1.2 Thin plate (a/h=100)

SS-AO3

FSDTN

orm

alis

ed r

esul

ts

w σx σy τxz τyz τxy0

0.2

0.4

0.6

0.8

1

1.2Moderately thick plate (a/h=10)

SS-AO3

FSDTN

orm

alis

ed r

esul

ts

Numerical Example 1

Sandwich plate under bi-directional sinusoidal transverse loading

0.0 3.0 6.0 9.00.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

20.0SS-AO3 (0/90/0), ASS-AO3 (90/0/90), ASS-AO3 (-30/60/-30), ASS-AO3 (45/-45/45), ASOLSH190 (0/90/0), ASOLSH190 (90/0/90), A

Displacement

Tra

nsve

rse

shea

r lo

adin

g q

Annular sandwich plate under uniform transverse shear at one end

Numerical Example 2

α

0.0 3.0 6.0 9.00.0

5.0

10.0

15.0

20.0

25.0

30.0SS-AO3 (0/0/0), ASS-AO3 (0/0/0), BFSDT-AO3 (0/0/0), AFSDT-AO3 (0/0/0), BSOLSH190 (0/0/0), ASOLSH190 (0/0/0), B

Displacement

Tra

nsve

rse

shea

r lo

adin

g q

Numerical Example 3

Cylindrical sandwich shell under point load

0.0 0.3 0.6 0.9 1.20

5000

10000

15000

20000

25000

SS-AO3 (4x4)

FSDT-AO3 (4x4)

BK20 (32x32x6)

Displacement

For

ce P

(10

3)

Summary

Formulation of 9-node sandwich shell element

1 Displacement fields

2 Transverse shear stress

3 Co-rotational framework

4 Shell coordinate system

5 Layer-wise theory

Thank you!

Questions?