# Presentation: 222819

## Comparative Analysis of Layered Elastic Computer Programs – LEAF, JULEA, and BISAR

- Gordon F. Hayhoe
- FAA AAR-410

- 4/20/04

## Need for Verifying the Accuracy of LEA Programs

- LEA programs work well for simple structures.
- LEA programs can give very low accuracy solutions under certain conditions.
- Very few closed form solutions which can be used to provide numerical checks.
- Compare results from multiple programs and select the “correct” answers by consensus and reasonableness.

- 4/20/04

## Critical Conditions for Pavement Design

| Condition | Cause of Inaccuracy | Application Affected |
| --- | --- | --- |
| Evaluation points close to the top of the structure | Gibbs wiggles | FWD backcalculation |
| Very thin layers | ? | Avoid with minimum thicknesses |
| Multiple fully unbonded interfaces | Ill-conditioned equations | Unbonded overlays |
| Evaluation points a large distance from the origin | Small numbers and ill-conditioning | Multiple-gear aircraft, B-747 and A380 |

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## Layered Elastic Equations

- is vertical stress, for example, where:
- equation is inverse Hankel transform.
*q* = tire pressure
*a* = tire radius
*r* = evaluation point radius
*z* = depth from top of layer
-  ** = Hankel domain variable

*r*

*z*

*h **i*

*a*

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## Vertical Strain – Summation by Gauss-Laguerre

- 4/20/04

## Evaluation Points Close to the Top of the Structure

- Load is constant pressure and discontinuous on the surface at the edge of the contact area.
- Therefore, stresses are discontinuous on the surface.
- Computed values close to the surface are unreliable because discontinuous functions cannot be properly represented by discrete transforms (using numerical integration).

- 4/20/04

## Different Methods

- BISAR changes to a special set of infinite series equations valid when the evaluation point is close to the surface.
- JULEA aborts stress calculation when the evaluation point is close to the surface. JULEA extrapolates for vertical deflection (a special case).
- LEAF shifts the origin vertically – improves the accuracy but does not eliminate the problem completely for stresses.

- 4/20/04

## Different Methods

- BISAR has two equation solvers, user selected. One is for bonded interfaces and the other is for unbonded interfaces.
- LEAF has three equation solvers, automatically selected (on magnitude of residuals).
- Part inverting – fastest, but inaccurate with ill-conditioned equations.
- Partial pivoting (LU) – intermediate.
- Full pivoting (Gauss-Jordan) – slow, but appears to give satisfactory accuracy under all critical conditions identified and tested.

- 4/20/04

## Comparison with Boussinesq for a Uniform Half-Space

*z**eval* = distance from surface

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## Comparison with Boussinesq for a Uniform Half-Space

*z**eval* = distance from surface

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## Surface Stresses

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## Surface Stresses

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## Vertical Stress Variation with Depth

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## Horizontal Stress Variation with Depth

- BISAR output precision is three significant figures in scientific format.

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## Horizontal Stress Variation with Depth

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## Horizontal Stress Variation with Depth

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## Widely Spaced Evaluation Points

- LEDFAA 1.3 computes the vertical subgrade strain in flexible structures with the contributions from all wheels in the main gear of the aircraft.
- This is 16 wheels for the B-747 and 20 wheels for the A380.
- Need to be sure that the contributions of the wheels far away from the evaluation point are accurate and reliable.

- 4/20/04

## Widely Spaced Evaluation Points

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## Widely Spaced Evaluation Points

- The contributions of the wheels far away are very small and a cut off distance could be used in many cases.
- But an appropriate cut off radius would vary with structure type, thickness, subgrade strength, etc.
- Therefore, use a consistent method.

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## Widely Spaced Evaluation Points

- Flexible structure.
- 5 in surface, 8 in stabilized base, 11 in subbase, 10 CBR subgrade.
- 50,000 lb single-wheel load.
- Seven evaluation points, 100 in intervals.

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## Vertical Strain at Top of Subgrade

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## Vertical Strain at Top of Subgrade - Expanded

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## Vertical Strain at Top of Subgrade – Expanded More

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## Transverse Stress at Top of Subgrade – Expanded Scale

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## Radial Stress at Top of Subgrade – Expanded Scale

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## Vertical Stress at Top of Subgrade – Expanded Scale

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## Conclusions

- It is important to check the accuracy of LEA programs for extreme conditions.
- The latest version of LEAF, used in LEDFAA and BAKFAA, provides accurate and reliable results for:
- FWD backcalculation.
- Unbonded concrete overlays.
- Multiple-gear aircraft having widely spaced main gear wheels.

- 4/20/04