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Geometrically Nonlinear Geometrically Nonlinear Finite Element Analysis Finite Element Analysis ofof
a Composite Reflector a Composite Reflector
K.J. Lee, G.V. Clarke, S.W. Lee, and K. Segal
FEMCI WORKSHOP
May 17 , 2001
MotivationMotivation• Recognized a need for nonlinear FEA in our
branch
• Interested in Element Design, Element Locking, and the effects of element distortion on solution accuracy
• Found a suitable project with the appropriate resources (interest, money, and time)
ObjectiveObjective
Develop a finite element research model that can be used to study highly flexible structures undergoing geometrically nonlinear behavior
Use the model to predict the structural response of a thin composite reflector
The ReflectorThe Reflector
Problem Formulation HighlightsProblem Formulation Highlights
ApproachApproach
Compute the deformation of a composite reflector with an areal density of 4 kg/m2 under point loads using a geometrically nonlinear solid shell finite element model.
Validate the computational results through comparison with the experimental data.
Problem Formulation HighlightsProblem Formulation Highlights
The Solid Shell ApproachThe Solid Shell Approach
Elements
4-node plate elements (6 dof/node) 9-node shell elements (6 dof/node)18-node shell elements ( 3-dof/node)
Solid Shell FormulationSolid Shell FormulationAll kinematic variables : vectors only
No rotational angle are used : easy connections between substructures
Problem Formulation HighlightsProblem Formulation Highlights
Element Locking: Zero-Energy Element Locking: Zero-Energy (Spurious) Modes(Spurious) Modes
Plate and shell elements lock as the thicknesses decrease
METHODS OF ELIMINATION– Reduced/ Selective Integration– Stabilization Matrix – Assumed Strain approach
Problem Formulation HighlightsProblem Formulation Highlights
Assumed Strain FunctionAssumed Strain Function
P ( , )where
H G q e1
and
H P C P dv
G P C B dv
v
Te
v
Te
Problem Formulation HighlightsProblem Formulation Highlights
Bubble DisplacementBubble Displacement
Eliminates some forms of element lockingReduces the sensitivity of elements to
distortion
Problem Formulation HighlightsProblem Formulation Highlights
Bubble Displacement FunctionBubble Displacement Function
element ralquadrilate node-9for )1)(1(N
element ralquadrilate node-4for )1)(1(
22b
22
bN
Problem Formulation HighlightsProblem Formulation Highlights
Model EquationsModel Equations
0 WdvCv e
T
HELLINGER-REISSNER FUNCTIONAL
EQUILIBRIUM
WdvCC eT
v eT
R )2
1(
0)( dvCv e
T COMPATIBILITY
The a3 vectors
Problem Formulation HighlightsProblem Formulation Highlights
Finite Element EquationsFinite Element Equations
))(,()2
(),())(,(
FUNCTION BUBBLE WITHNTDISPLACEME ASSUMED
)2
)(,())(,(
GEOMETRY
01
01
301
zbbbiz
n
iii
n
ii
iii
n
ii
uuNut
NuNu
at
NxNx
Problem Formulation HighlightsProblem Formulation Highlights
Element Stiffness MatrixElement Stiffness Matrix
GHGK Te
1
Problem Formulation HighlightsProblem Formulation Highlights
Nonlinear EquationsNonlinear Equations
0)FF(qKint
)()( iext
i
Δqqq (i)1)(i
ε)1(
q
qifConverged
i
Geometry Of the Reflector
A very thin structure
The modeled reflector The modeled reflector (bottom view)(bottom view)
Stiffener (Frame)Stiffener (Frame)
Material Properties and Lay-upsMaterial Properties and Lay-ups
M60J/954-3 Unitape E1: 53 Msi (Tensile), 50 Msi (Compressive)
E2, E3: 0.95 Msi (Tensile).0.91 Msi (Compressive)
G12, G13: 0.0681 Msi, G23: 0.3185 Msi
ν12 , ν13 :0.319, ν23: 0.46
Layups Dish shell: (0/90/45/135)s – 8 plies, 0.016" thick
Frame/Cap: (0/60/-60)s2 – 12 plies, 0.024" thick
Load CasesLoad Cases
Three cases of point load applied on the reflector surface
Load directionLoad direction
The direction of load is fixedAlways perpendicular to the original
reflector surface
Boundary ConditionsBoundary Conditions
Case 1 –ResultsCase 1 –Results
0
2
4
6
8
0 0.2 0.4 0.6 0.8 1 1.2
Z-Displacement [inches]
Lo
ad
[lb
s]
Test
Linear
Nonlinear
Case 1 - Deformed ShapeCase 1 - Deformed Shape
Case 2 - ResultsCase 2 - Results
0
2
4
6
8
10
0 0.2 0.4 0.6 0.8 1
Z Displacement [inches]
Lo
ad
[lb
s]
Test
Linear
Nonlinear
Case 2 - Deformed ShapeCase 2 - Deformed Shape
Case 3 - ResultsCase 3 - Results
0
2
4
6
8
0 0.1 0.2 0.3 0.4 0.5
Z Displacement [inches]
Lo
ad
[lb
s]
Test
Nonlinear
Case 3 - Deformed ShapeCase 3 - Deformed Shape
ConclusionConclusion
The computed displacements are in reasonably good agreement with the experimental results.
The finite element software, based on the assumed strain solid shell formulation, can be used for analysis of highly flexible space structures such as a composite reflector.
Current and Future WorkCurrent and Future Work
User friendly program: Combine with the existing pre or post-processing program (e.g., FEMAP)
Geometrically nonlinear analysis under dynamic loading during launch
Implementation of a triangular element for ease of mesh design and refinement
Why triangular elements ?Why triangular elements ?