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 ?