# Presentation: 106491

## Using Plate Elements for Modeling Fillets in Design, Optimization, and Dynamic Analysis

- FEMCI Workshop, May 2003
- Dr. Andrew M. Brown
- ED21/Structural Dynamics & Loads Group
- NASA 	Marshall Space Flight Center

## Motivation

- Many structures best modeled with plate elements
- More accurate for “plate-like” structures
- Significantly less computationally intensive – still important for optimization, dynamic analysis, small business

- Presence of fillets, however, requires solid modelling
- Automeshing thin-walled structures with solids problematic
- Manually meshing fillets with solids not trivial

- Can ignore fillet frequently, but *not always*
- Significantly underpredict stiffness for thin-walled structures

## Solution & Methodology

**Goal**: Generate simplified technique for modeling fillets for use in design, dynamic analysis.
- Literature search performed on analytical methods for natural frequencies of plates with variable cross section.

- Evaluated several options
- NASTRAN “GENEL” explicit element
- “stepped-up thickness” method

****

- Element 2 (“plate”): Thickness t, Length L

- Element 1 (bridge): Thickness tb, Length Lb

- Element 3 (bridge): Thickness tb, Length Lb

- “wall”
- thickness twall

- “Colinear” non-filleted section

- Node 1

- Node 2

- Node 3

- Node 4

- Decided on “Bridge Element” method.

## Theoretical Development - Bridge

- First recognize that “wide-beam” theory allow use of beam equations for plate deflection under plane strain:
- Generate Stiffness matrix partition for node 4 of bridge configuration:
- Equation for rotation at node 4 in form of coefficients of load P and moment M in terms of unknowns tb and Eb:

## Theoretical Development – plane fillet

- For actual fillet, express rotation at tangent point as
- Where
- Applying boundary condition, obtain nonlinear expression for rotation at tangent:

## Equations for Eb and tb

- Equate the coefficients of P & M in each expression for f

- Solve for tb/t and Eb/E using MathematicaTM 4.1

## Results

- Surface fit for each generated to fit data for easy implementation:
- where x=r/t and y=twall/t
- Matrix of exact results also generated for Eb/E= f(r/t,tw/t),  tb/t= f(r/t,tw/t).
- For accurate dynamic analysis, pseudo density of bridges also calculated:

## Validation – Sub-Component Level

- “Wide-beam” and plane-strain assumptions validated with f.e. models.
- Fillet tangent rotation equation using beam theory compared with f.e.; 7.8% error

## Validation – Structural Level

- Representative thin-walled structure fabricated, modeled, modal tested.
- Dense solid model, plate model with “bridge” fillets, plate model ignoring fillets altogether.

## Excellent Correlation, 90% DOF Reduction

- Mode shapes consistent
- Error of “bridge” plate model same as solid model
- Error of plate model ignoring fillets substantial.
- Solid model - 257,600 dofs
- Plate “bridge” model 22,325 dofs

## Conclusions & Future Work

- Methodology developed that uses plate instead of solid elements for modeling structures with 90o fillets
- Technique uses plate “bridges” with pseudo Young’s Modulus, thickness, density.
- Verified on typical filleted structure
- accuracy better than or equal to a high-fidelity solid model
- 90% reduction in dof’s.
- Application of method for parametric design studies, optimization, dynamic analysis, preliminary stress analysis
- Future work: extend theory to fillet angles other than 90°, multi-faceted intersections.