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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
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Background
Sandwich structures in civil engineering:
Insulation walls Roof panels Curtain wall glazing
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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
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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( , )
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• Modulus ratio
Transverse Shear Stress Through Thickness
R=105 R=106R=102R=10
face
core
ER=
E
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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)
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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
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Additional freedoms are defined in local shell coordinate system.
Shell Coordinate System
xi yi( , )
z xyn
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Additional freedoms are defined in local shell coordinate system.
Shell Coordinate System
xi yi( , )
nz
x
y
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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.
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Layer-wise Theory
Hellinger-Reissner Variation Principle
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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
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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
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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)
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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
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Thank you!
Questions?