Gordon F. Hayhoe FAA AAR-410 Comparative Analysis of Layered Elastic Computer Programs – LEAF, JULEA, and BISAR Gordon F. Hayhoe FAA AAR-410 4/20/04 1

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 2

Critical Conditions for Pavement Design 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 4/20/04 3

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 a = Hankel domain variable a r z h i 4/20/04 4

Vertical Strain – Summation by Gauss-Laguerre 4/20/04 5

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 6

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 7

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 8

Comparison with Boussinesq for a Uniform Half-Space zeval = distance from surface 4/20/04 9

Comparison with Boussinesq for a Uniform Half-Space zeval = distance from surface 4/20/04 10

Surface Stresses 4/20/04 11

Surface Stresses 4/20/04 12

Vertical Stress Variation with Depth 4/20/04 13

Horizontal Stress Variation with Depth BISAR output precision is three significant figures in scientific format. 4/20/04 14

Horizontal Stress Variation with Depth 4/20/04 15

Horizontal Stress Variation with Depth 4/20/04 16

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 17

Widely Spaced Evaluation Points 4/20/04 18

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. 4/20/04 19

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. 4/20/04 20

Vertical Strain at Top of Subgrade 4/20/04 21

Vertical Strain at Top of Subgrade - Expanded 4/20/04 22

Vertical Strain at Top of Subgrade – Expanded More 4/20/04 23

Transverse Stress at Top of Subgrade – Expanded Scale 4/20/04 24

Radial Stress at Top of Subgrade – Expanded Scale 4/20/04 25

Vertical Stress at Top of Subgrade – Expanded Scale 4/20/04 26

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 27