# Presentation: 072887

## Analysis of Gossamer Space Structures Using Assumed Strain Formulation Solid Shell Elements

**Keejoo Lee and Sung W. Lee**
- Department of Aerospace Engineering
- University of Maryland College Park
**Gregory V. Clarke**
- Goddard Space Flight Center
**2002 FEMCI WORKSHOP**

## Motivation

- Because of zero-gravity environments involved in conducting experiments, greater emphasis may be placed on computational analysis of gossamer space structures.
- Length ( ~ 100 m)  to thickness (~1 micron) ratios, **L/t**, are very large (~ L/t=108).

## Objective

- The purpose of this work is to investigate the applicability of assumed strain formulation solid shell elements to gossamer space structures.
- Shell elements suffer from locking as thickness decreases – it is necessary to determine the range of L/t for which the assumed strain formulation solid shell elements perform with precision.

## Solid Shell Formulation

- Treats shells as 3-dimensional solids.
- No rotational angles are used.
- - All kinematic variables are vectorial.
- - All kinematic variables are based on a global coordinate system.
- - Allows transverse shear deformation.
- - Stretchable in the thickness direction.

## Assumed Strain Formulation

- _Assumed displacement formulation_
- 	 - Susceptible to element locking.
- _Assumed strain formulation_
- - In addition to the assumed displacements, assumed strain field, independent of the assumed displacements, introduced to avoid  element locking.
- 	 - Assumed strain parameters eliminated at element level.

## 54 DOF Elements

**18 Node Version**
**3 DOF/NODE**

**9 Node Version**
**6 DOF/NODE**

## Numerical Tests

- To investigate the applicability of the assumed strain formulation solid shell elements to extremely thin structures under the following conditions:
- 	 - Geometrically linear and nonlinear.
- 	 - Linear elastic materials.
- 	 - Static loadings	.

## The Effect of Element Locking

- Linear analysis of a beam under uniformly distributed load is carried out to appreciate the effect of element locking.

## Locking Tests

## Modified Moduli

- To increase the effective range of length/thickness ratio, one may introduce modified moduli  for the beam as follows:

## Excluding the contribution of membrane energy,the total strain energy U can be expressed as

**Excluding the contribution of membrane energy,****the total strain energy U can be expressed as**

## Similarly, one may introduce modified Young’s modulus,

**Similarly, one may introduce modified Young’s modulus,**
**corresponding to the thickness change as follows:**

## With Modified Moduli

## Quadruple Precision

- One may numerically extend the effective range of length-to-thickness ratios using the quadruple precision.
- 	- Double precision ~ 15 decimal digits.
- 	- Quadruple precision ~ 30 decimal digits.

## Geometrically Linear Analysis

## With No Modification to Moduli

## Modified Moduli

## With Modified Moduli

## Geometrically Nonlinear Analysis

## Wrinkling

- Gossamer structures are susceptible to wrinkling.
- To predict details of wrinkles such as shapes and amplitudes, one may use a dynamic scheme that includes the effect of mass and fictitious damping.

## A Square Membrane

- Material: Kapton

- P

- (1)

- (1)

- (2)

- (2)

- P

- Boundary Conditions
- -  (1):  *u*=0
- -  (2):  *v*=*w*=0

## Assumed Imperfection

**Notes:**
J. Arbocz, “ The imperfection data bank”

## Wrinkling Formation

## A Square Membrane

- P

- P

- 0.1P

- 0.1P

## Wrinkle Formation

## Observations (1)

-  From the numerical tests, **Max. L/t ranges** are
-       - Double precision w/o modification 	~ 105
- 	  - Double precision with modification 	~ 108
-  	   - Quadruple precision w/o modification  ~ 1016
- 	   - Quadruple precision with modification ~ 1018

## Observations (2)

- The use of quadruple precision, combined with the modified moduli significantly increases the effective range of length-to-thickness ratios for assumed strain solid shell elements.
- Also, it appears that the wrinkling formation of membranes can be simulated via a geometrically nonlinear analysis, incorporating the fictitious damping and initial geometrical imperfections.