# Presentation: 090140

## The Use Of Non-standard Devices In Finite Element Analysis 

- Willi W. Schur, Ph.D.
- Physical Science Laboratory-New Mexico State University
- Field Engineering Group
- Attached to the
- NASA Balloon Programs Office
- GSFC-Wallops Flight Facility
- Wallops Island, VA 23337

## Membrane & Cable Structures

- Are able to undergo large deformations while remaining in the small strain regime.
-      * geometrically non-linear problem
- 	-- analyses by digital methods require an evolutionary
-          process
- Are generally under-constrained.
-     * FE modeling leads to a singular stiffness matrix.
-     -- FEA by an implicit solution scheme is prevented.
- Membranes and cables are uni-directional (no compression).
-     * Slackness of a structural element may have to be dealt
-        with in the analysis.

## Under-constrained structures are common in technology applications

- Assessment of the structural response of such systems is important.
- The analyst must find ways to assess such systems.
- Practical designs generally have the uniqueness property;
-      i.e. the system has destiny.
-      * There is only one solution for a loading state; this solution is independent of the loading path (evolution).
- Systems that have several solutions do not only pose problems to analysis but also to deployment.

## FEM processes that can aid analyses of under-constrained structures

- The dynamic response of a structural system with discrete masses can be analyzed without stiffness matrix inversion using an explicit solution scheme.
-      * Requires sufficiently small time steps so that a signal cannot overstep an element.
- Artificial constraints can be used to guide the evolution to the loaded state.
-      * Ideally these constraints should be diminished or removed  towards the end of the evolution.
- In a non-linear FEA, the evolution to the loaded state can be started by displacement of the boundary to induce self-stress such that the tangent stiffness matrix for the next step has a geometric component that renders the stiffness matrix non-singular.
-     * some sufficiently simple problems only

## Pseudo-dynamic processes for static problems (No matrix inversion)

- Dynamic Relaxation
-     * Arbitrary lumped masses and lumped viscous damping are applied to a static problem.
-     * Masses and Dashpots are chosen to make the system nearly critically damped.
-      * An explicit solution scheme is used to march through the transient.  The post transient solution is the static solution.
-     -- Time step size limitations apply to assure stability of the solution process.  (Well established in the literature.)
- Evolution using lumped damping alone should work similar to Dynamic Relaxation.

## Use of Artificial Constraints in Membrane Structures

- From  1st Order Thin Shell Theory it is known that the membrane solution is essentially decoupled from the bending solution.
-     * provided in the early 20th century justification to use membrane analysis prediction for the interior field of a thin shell.
-      * provides justification to supply small artificial bending stiffness to a membrane to enable and ease analysis of a membrane mechanics problem.
-      -- there is a need to provide checks for validity

## Other Methods to Deal With Under-constrained Structures 

- One can write constraint equations for inelastic structures.
-     * typically non-square matrices are obtained
-     -- over-constrained systems imply the possibility of self-stress
-     -- under-constrained systems are mobile
-     * useful methods involve a pseudo-inverse of a constraint matrix
- Practical methods seem to be limited to structures with one dimensional elements.
- The inclusion of elastic response makes these methods unwieldy.

## Slackness of Cables and Wrinkling of Membranes (approximate techniques)

- Slackness of elastic cables can be approximated using a bi-linear representation of the strain response behavior.
- The in-plane component of membrane wrinkling can be modeled by a tension field.
-      *Tension Field (TF) response can be implemented by a non-  linear material model.
-      *TF response ignores the minutia of the wrinkles.
-      *Penalty parameter formulation avoids degradation of non-singular status of the stiffness matrix.
-      *Single integration point reduced integration membrane elements avoid toggling at the tension field boundary

## Additional Comments on Wrinkling

- TF response cannot provide details on wrinkles.
- Some progress has been made on characterizing wrinkles in a TF.
- In some cases, the details of the wrinkles are determined by what happens at the boundary of the TF rather than in its interior.

## Structural Lack of Fit

- Structural lack of fit can be used to obtain more efficient structural systems.
- In solid structures, structural lack of fit requires force fit at assembly/fabrication.
- Tension only structural elements can be assembled with lack of fit without requiring force fit.

## FEA Analyses With Lack of Fit

- In solid structures and some special membrane structures lack of fit is easily implemented in a multi-step solution process; in most membrane structures it is not.
- Structural lack of fit can also be modeled analytically by non-linear material models.

## Inclusion of Initial Slackness in Membrane Structures 

- Membranes with initial slackness
-     * In a FE code material excess can be modeled in the material (constitutive law) module.
-     * The material module must have Tension Field capability

## Pumpkin Shape Super-pressure BalloonUse of Non-standard Devices

- Small artificial bending stiffness
-      *Large enough to overcome numerical difficulties
-      *Small enough to only negligibly affect membrane solution
- TF to allow wrinkling of skin
- Material excess and lack of fit to obtain favorable strength requirements for design critical state

## Pumpkin Shape Super-pressure BalloonParametric Study With Film Properties Fixed

## Summary of Ideas

- Commercial General Purpose FE Codes provide an arsenal of usually proven tools for large classes of structural systems.
- There are analysis needs that cannot be met using proven commercial codes alone.
- Some of these needs can be met by cautious use of non-standard devices.
- Particular care must be taken to legitimize the use of such devices (possibly on a case by case basis).
- The use of Penalty parameters with the FEM is well established.  It’s extension to TF modeling and other features is as legitimate as it’s use for example in contact problems.