# Presentation: 226798

## Geometrically Nonlinear Finite Element Analysis of a Composite Reflector

**K.J. Lee, G.V. Clarke, S.W. Lee, and K. Segal**
**FEMCI WORKSHOP**
**May 17 , 2001**

## Motivation

- 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)

## Objective

- 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 Reflector

## Problem Formulation HighlightsApproach

- 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 HighlightsThe 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 Formulation

- All kinematic variables : vectors only
- No rotational angle are used : easy connections 					       between substructures

## Problem Formulation HighlightsElement Locking:  Zero-Energy (Spurious) Modes

- Plate and shell elements lock as the thicknesses decrease
- METHODS OF ELIMINATION
- Reduced/ Selective Integration
- Stabilization Matrix
- Assumed Strain approach

## Problem Formulation HighlightsAssumed Strain Function

- where

- and

## Problem Formulation HighlightsBubble Displacement

- Eliminates some forms of element locking
- Reduces the sensitivity of elements to distortion

## Problem Formulation HighlightsBubble Displacement Function

## Problem Formulation HighlightsModel Equations

- HELLINGER-REISSNER FUNCTIONAL

- EQUILIBRIUM

- COMPATIBILITY

## The a3 vectors 

- The a3 vectors

## Problem Formulation HighlightsFinite Element Equations

## Problem Formulation HighlightsElement Stiffness Matrix

## Problem Formulation HighlightsNonlinear Equations

- Geometry Of the Reflector

-  	A  very thin structure

## The modeled reflector (bottom view)

- The modeled reflector (bottom view)

## Stiffener (Frame)

## Material Properties and Lay-ups

**M60J/954-3 Unitape**
-    		**E****1****: 53 Msi (Tensile), 50 Msi (Compressive)**
**   ****	****	****E****2****, E****3****: 0.95 Msi (Tensile).0.91 Msi (Compressive)**
**   ****	****	****G****12****, G****13****: 0.0681 Msi,  G****23****: 0.3185 Msi**
**   ****	****	****ν****12 ****, ****ν****13**** ****:0.319, **** ν****23****: 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 Cases

- Three cases of point load applied on the reflector surface

## Load direction

- The direction of load is fixed
- Always perpendicular to the original reflector surface

## Boundary Conditions

## Case 1 –Results

## Case 1 - Deformed Shape

## Case 2 - Results

## Case 2 - Deformed Shape

## Case 3 - Results

## Case 3 - Deformed Shape

## Conclusion

- 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 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 ?