Comparative Analysis of Layered Elastic Computer Programs

1
Comparative Analysis of Layered Elastic Computer
Programs LEAF, JULEA, and BISAR

• Gordon F. Hayhoe
• FAA AAR-410

2
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.

3
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
4
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
5
Vertical Strain Summation by Gauss-Laguerre
6
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).

7
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.

8
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.

9
Comparison with Boussinesq for a Uniform
Half-Space
zeval distance from surface
10
Comparison with Boussinesq for a Uniform
Half-Space
zeval distance from surface
11
Surface Stresses
12
Surface Stresses
13
Vertical Stress Variation with Depth
14
Horizontal Stress Variation with Depth
BISAR output precision is three significant
figures in scientific format.
15
Horizontal Stress Variation with Depth
16
Horizontal Stress Variation with Depth
17
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.

18
Widely Spaced Evaluation Points
19
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.

20
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.

21
Vertical Strain at Top of Subgrade
22
Vertical Strain at Top of Subgrade - Expanded
23
Vertical Strain at Top of Subgrade Expanded More
24
Transverse Stress at Top of Subgrade Expanded
Scale
25
Radial Stress at Top of Subgrade Expanded Scale
26
Vertical Stress at Top of Subgrade Expanded
Scale
27
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.