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Published in Third Int. Conf. on Road and Airfield Pavement Technology, Beijing, April 1998.

 

RECENT DEVELOPMENTS IN FLEXIBLE AIRCRAFT PAVEMENT DESIGN
USING THE LAYERED ELASTIC METHOD

Leigh J. Wardle
Director, MINCAD Systems Pty. Ltd.
Australia

Bruce Rodway

Chief Engineer- Pavements, Federal Airports Corporation
Australia

 

ABSTRACT

The conventional empirical methods for structural design of flexible aircraft pavements were adapted from highway practice, then modified and extrapolated to cater for the airfield situation. The methodology was calibrated against the US Army Corps of Engineers full-scale trafficking tests conducted over 25 years ago. These conventional design methods are now generally recognised to be inadequate to assess the effect of Boeing’s B777, and of proposed New Large Aircraft (NLA). The B777’s 6-wheel tridem gears and the NLA gears were not represented in the Corps tests. Also the conventional methods use single layer analysis so have no direct mechanism for crediting bound layers for their superior load spreading characteristics.

The layered elastic method was introduced into design practice in the mid-1990’s, with the release of the computer program LEDFAA by the U.S. Federal Aviation Administration and also the Australian-developed program APSDS (Airport Pavement Structural Design System). These two ‘mechanistic’ methods differ in a number of respects. Design case studies are analysed with APSDS and comparisons of results made with LEDFAA. Reasons for differences in the results are discussed. Additional examples are provided to demonstrate the significance of the degree of wander in determining the required thicknesses of runways relative to taxiways and docking bays.

ICAO’s Aircraft Classification Number (ACN) indicates an aircraft’s pavement damaging effect relative to other aircraft. The rating system uses the conventional empirical design method and predicts damage by the B777 and NLAs that appears to be unrealistically high for low strength subgrades relative to that caused by the B747. This situation is not resolved, however, by the introduction of ‘mechanistic’ methods based on layered elastic analysis. Both LEDFAA and APSDS are calibrated against the same limited full-scale tests used to produce the empirical design method and so predict similar B777 pavement damage. Because the pavement design methods remain essentially empirical, confident assessment of the damage by new significantly different aircraft wheel arrangements await the completion of appropriate full-scale tests of the kind programmed for FAA’s National Airport Pavement Test Facility now under construction at Atlantic City. A recent rutting test for Boeing in which B777 and B747 gears produced equal pavement deformation confirms the need for further full-scale testing.

INTRODUCTION

Most current procedures for flexible airport pavement design and analysis, in particular the Federal Aviation Administration (FAA) design method and the International Civil Aviation Authority (ICAO) system for aircraft load classification, have their origins in the empirical CBR thickness method originally developed for highways. The procedures involve simplifying assumptions that were necessary for manual calculation, but which are no longer justified given the availability of desk-top computers. The major simplifications are related to the use of single layer elastic theory, the pass-to-coverage ratio to address aircraft wander, and the deflection-based Equivalent Single Wheel Load (ESWL) concept for multiwheel aircraft gear.

The degree of channelisation of aircraft traffic is very different for runways, taxiways, docking bays, etc. This transverse spreading of load to different degrees due to aircraft ‘wander’ significantly affects the required pavement thickness. The statistics of aircraft wander were initially accounted for in terms of a pass-to-coverage ratio (PCR). A point on the pavement is said to receive a ‘coverage’ when any part of a tyre’s contact area passes over it. The PCR is defined as the number of passes of a wandering aircraft that is statistically required for the most frequently covered point to receive one coverage. The PCR depends upon wheel configuration, tyre width and the degree of wander. The original PCR concept solely addresses the statistics of load distribution at the pavement surface and, therefore, incorrectly implies that the reduction in pavement damage due to aircraft wander is the same for all pavement thicknesses.

The empirical CBR method of pavement thickness design relates ESWL, pavement thickness and subgrade CBR. It uses single layer analysis so has no direct mechanism for crediting bound layers for their superior load spreading characteristics. Thick bound layers are increasingly used, however, and are required by airport authorities such as the FAA. Bound layers are typically accounted for within the empirical design by using layer equivalency factors based largely on elastic layer theory. The CBR method assumes a failure mode that consists of surface deformation (rutting) caused by overstressing the subgrade. Pavement failure due to fatigue cracking of the bituminous surfacing or cracking of other bound layers is not addressed by the method and must be separately considered by the designer. Although this paper primarily deals with the application of layered elastic design methods to the surface deformation failure mode, they are also applicable to fatigue cracking of bound layers.

A validated ‘rational’ method of pavement analysis is needed primarily to cater for bound layers and to assess more rigorously the degree to which wheels and wheel groups interact to cause increased pavement damage. The need has become more urgent with the arrival of the B777 whose 6-wheel gear configuration differs from those used in the full-scale pavement tests. The urgency is increased by plans for very large aircraft with more wheels on each gear and with close gear spacings.

Efforts by researchers have focussed on the layered elastic method as that most likely to provide an adequate solution within the short time frame that is available. Elastic models with isotropic (same property values along all axes) layers have been used to compute maximum values of chosen damage indicators, most commonly subgrade strain, which are then related to pavement life (strain repetitions) by calibrating against full-scale trafficking tests. This calibration process seeks to faithfully translate the test pavement behaviour to the design and analysis of new pavements. These commonly consist of materials and thicknesses that are different from the test pavements, and are trafficked by aircraft whose wheel configurations differ from those used in the tests. The same elastic properties assumed for the test track materials for calibration purposes must be assigned to the same materials if they appear in the new pavements. Failure to do this fractures the vital empirical connection between the reality of the test track and the behaviour of the new pavements. Without this empirical anchor, the design is adrift and unsubstantiated. Application of the design method to pavements containing new materials involves assigning realistic elastic stiffnesses to those materials. Judgements are typically based on laboratory tests such as triaxial tests or moduli inferred from field deflection tests.

The layered elastic method was introduced into airfield design practice in the mid-1990’s, with the release of the computer program LEDFAA (Layered Elastic Design, Federal Aviation Administration) and also the Australian-developed APSDS (Airport Pavement Structural Design System), (Federal Aviation Administration, 1995) and (Wardle and Rodway, 1995) respectively.

APSDS

APSDS is based on a concept described by Monismith et al. (1987). The important new and unique feature is that subgrade strains, or alternative indicators of the rate at which deformation develops at the pavement surface, are computed for all points across the pavement in order to capture all damage contributions from all the aircraft wheels in all their wandering positions. This contrasts with previous methods which computed single maximum values of the damage indicators. It is this feature that eliminates the need for the pass-to-coverage concept and allows the designer to specify any degree of wander. Successive aircraft movements have been observed to be normally distributed about the pavement centreline. The standard deviation (SD) for a taxiway is typically taken as 773 mm and for a runway as 1546 mm (Ho Sang, 1975). These correspond to wander widths of 1778 mm (70 inches) and 3556 mm (140 inches) where wander width is defined by the zone containing 75% of the aircraft centrelines. For a docking bay, a SD of the order of 200 mm may be appropriate.

In the case of fatigue failure of bound layers, APSDS would typically compute the tensile strains at the undersides of the relevant layers. Strains are computed using the layered elastic program CIRCLY (Wardle, 1977).

The strains are converted to damage using a performance relationship of the form:

(1)

where N is the predicted life (repetitions of e )

k is a material constant

b is the damage exponent of the material

e is the load-induced strain (unitless strain)

More complex performance relationships could be accommodated by the program if required.

The Damage Factor for the i-th loading is defined as the number of repetitions (n_i) of a given damage indicator divided by the ‘allowable’ repetitions (N_i) of the damage indicator that would cause failure. The Cumulative Damage Factor (CDF) is given by summing the damage factors over all the loadings in the traffic spectrum using Miner’s hypothesis:

Cumulative Damage Factor = (2)

The pavement is presumed to have reached its design life when the cumulative damage reaches 1.0. In the case of a surface deformation failure mode, parameters k and b must be established from full-scale aircraft trafficking tests to failure, using an iterative procedure as discussed below. For the fatigue mode, k and b can potentially be obtained from laboratory testing.

Calibration involves determining the values of parameters k and b for Equation 1 which best reflect the rutting behaviour of the Corps’ test pavements. The following example explains the method. Six tests, with B747, C5A and B36 test rigs trafficking pavements of different thickness to failure were used in the calibration process.

Test 3B-5 involved trafficking a twin tandem test rig (a single B747 gear) on a 1040 mm thick pavement over a CBR 4.0 subgrade. The pavement consisted of a 75 mm asphalt surface course over a 150 mm crushed rock basecourse over 815 mm uncrushed gravel subbase. Realistic modulus and Poisson’s ratios were assigned to the pavement materials and subgrade. The basecourse and sub-base were divided into sublayers and modulus values assigned to the sublayers in accordance with the Barker and Brabston methodology (Barker and Brabston, 1975). Full details of the twin tandem rig (wheel coordinates, load and tyre pressure) were included in APSDS.

Figure 1:Re-analysis of full-scale test showing damage contributions from individual lanes and total
(B747 dual tandem on 1040 mm pavement)

The test rig had travelled along five equally spaced guidelines with the number of passes for each guideline chosen to approximate a prescribed aircraft wander distribution. The pavement was deemed to have failed after a total of 454 passes (ie CDF=1.00). APSDS calculates and sums the damage contributions from the passes on each guideline as shown on Figure 1. The particular parameters k (= 0.004276) and b (= 6.635) used in the rutting damage model (Eqn 1) were iteratively determined to give a least squares best fit to the data from the six full-scale tests considered.

The APSDS user can specify all inputs, including material properties, wander, aircraft loadings and the damage model. This easy access and flexibility is intended to enable experienced pavement designers to utilize the program to give customized treatment to particular pavement situations. For example, within a localised area, perhaps as localised as a single airport, quality pavement performance data, including good estimates of past traffic may be available. In such a situation the designer could use the data to calibrate APSDS to reflect the local pavement experience. The tool is then available to readily compute the effects of future aircraft traffic and the use of new pavement materials.

Figure 2 is a typical cumulative damage ‘profile’ across the pavement generated by APSDS.

Figure 2 Sample cumulative damage plot

LEDFAA

LEDFAA is now an FAA standard design method (FAA Advisory Circular 150/5320-16 of October 1995, which includes the LEDFAA User’s Manual) and is used in parallel with FAA’s traditional empirical CBR design method (FAA Advisory Circular 150/5320-6D of July 1995). FAA currently restricts the use of LEDFAA to aircraft mixes containing at least some B777s. It is generally less transparent than APSDS and the user has less freedom to make basic changes to the program’s operation. This is to be expected given that the program functions as an FAA design standard.

APSDS has been calibrated against the original Corps of Engineers test data as described earlier. LEDFAA is not calibrated directly against the Corps data, but is conditioned by mandating certain input material properties to produce, for typical aircraft traffic mixes, similar pavement thicknesses to those obtained using the FAA conventional design method (McQueen et al, 1997). For example, in order to better align the LEDFAA and FAA conventional thicknesses, asphalt surfacing is assigned a constant stiffness of 1380 MPa which is low for many cooler environments, especially for thick asphalt layers. FAA acknowledge that the conditioning of LEDFAA is a transitional measure to facilitate the smooth introduction of mechanistic design methods to pavement design practice. It is expected that the mandated modulus values will be modified over time towards more ‘realistic’ values as performance data becomes available, and as better methods are developed for determining material properties. At the present time, changes made to the mandated inputs renders the design ‘non-standard’ for the purpose of FAA funding approvals and the reasonableness of the changes then need to be argued by the designer. LEDFAA produces pavements that are, on average, 3% thicker than the FAA conventional method. They are thicker for CBRs less than 5% and thinner for CBRs higher than 15% (McQueen et al, 1997).

LEDFAA processes traffic differently to the FAA conventional method. It retains the coverage concept but computes, for each aircraft, the number of coverages applied to each 10 inch wide strip of pavement. The lateral traffic distribution is fixed at taxiway wander and cannot be varied by the user. The total damage caused by all the aircraft expected to use the pavement is obtained by summing (using Miner’s Law) the damages for the critical strip. In this way LEDFAA recognizes that the landing gears of the various aircraft in the design traffic mix track along different paths relative to pavement centreline. This contrasts with the way in which the current FAA design method combines the effects of a mix of aircraft. The method requires that the departures of each aircraft be first converted into an equivalent number of departures of the ‘Design Aircraft’. The Design Aircraft is selected from the traffic mix as the aircraft which, due to its size and number of departures, would require the thickest pavement. The departures of all the other aircraft are then converted to equivalent departures of the Design Aircraft using a prescribed approximate method. The pavement is then designed for the total equivalent departures of the Design Aircraft only, using its pass-to-coverage ratio. This procedure was devised prior to the ready availability of computers in order to reduce the computational load. The FAA method in effect sums the maximum damage caused by each aircraft even though they track along different parts of the taxiway relative to the pavement centreline. This is a conservative procedure, to a degree that depends upon the composition of each traffic mix.

A further distinction between the conventional FAA and LEDFAA’s method of treating traffic is that the pass-to-coverage ratios are different. FAA’s are based on the overlap of tyre contact areas at the pavement surface whereas LEDFAA considers the overlapping effects of a wider ‘effective’ tyre width at subgrade level and also considers depth to subgrade relative to axle spacing when deciding the number of effective strain repetitions. These methodologies are described in Technical Manual TM 5-825-2-1/AFM 88-6 (US Army and Air Force, 1989). However, because LEDFAA computations are framed in terms of aircraft departures rather than coverages, the pass-to-coverage ratios used are not accessible to the user.

Thus LEDFAA processes the traffic in fundamentally different ways to those used by the FAA conventional design method. As stated earlier it is conditioned to produce similar pavement thicknesses for typical traffic mixes. Consequently the use of LEDFAA for single aircraft assessments is problematic and may produce conservative pavement thicknesses, as discussed in the LEDFAA manual. Nevertheless, LEDFAA can be utilised to indicate the relative effect of different aircraft as we show later for the B777 and B747. It should be noted that comparisons between design methods will be affected by the level of traffic volume at which the comparisons are made. This is because the effect of repetitions on required pavement thickness is not the same for each method. This is to be expected because there is little if any reliable data available for high coverage levels (the median level of test traffic to failure in the Corps’ full-scale tests was only 300 coverages with one test taken to 7000 coverages).

Subgrade compressive strain (termed ‘resilient’, ‘elastic’, or ‘recoverable’ strain) is often chosen as the indicator of the rate at which permanent, ‘plastic’ deformation accumulates, independent of subgrade stiffness. APSDS has to date adopted this model. The fatigue relationship used by LEDFAA was obtained by plotting the maximum strains against repetitions to failure for 37 tests (Barker and Gonzalez, 1991 and 1992). Unlike the current APSDS model, the LEDFAA relationship (see LEDFAA User Manual or Help file) indicates that stiffer subgrades give longer life for the same level of imposed subgrade strain.

Both APSDS and LEDFAA automatically sublayer unbound granular basecourse and subbases using the Barker and Brabston methodology (Barker and Brabston, 1975).

DESIGN AND EVALUATION EXAMPLES

In this section a number of design case studies for CBR 3 subgrades are developed with APSDS. First of all taxiway wander is assumed to allow comparison with LEDFAA. Some additional APSDS calculations are presented to demonstrate the significance of the degree of wander. Wander is a critical design parameter as it is significantly different for runways, taxiways and aircraft docking bays.

The sample design traffic mix used for illustration is:-

The subgrade CBR is assumed to be 3% (ICAO Subgrade ultra low strength category, Code D).

The following pavement structure is used:

The following table summarises the results:

	Total pavement thickness
Wander: Model	Taxiway (SD=780 mm)	Runway (SD=1560 mm)	Docking bay (SD=200 mm)
APSDS 3	2130 mm	2015 mm	2205 mm
LEDFAA 1.2	2145 mm	n/a	n/a

The pavement thicknesses for APSDS and LEDFAA are very similar. The APSDS results for alternative wander widths show that the pavement thickness can be reduced by about 5% for runway conditions. The pavement would need to be about 3.5% thicker for a docking bay. Note that the effect of wander would be more pronounced for higher strength subgrades because of the thinner pavements required. As mentioned above, LEDFAA is restricted to taxiway wander.

The predicted errors resulting from the LEDFAA assumption of taxiway wander can be expressed in terms of the pavement life calculated with APSDS using the alternative wander conditions. A docking bay pavement incorrectly designed using the taxiway wander would have a life of 12 years instead of the design life of 20 years. A runway pavement would be over-designed with it’s life extended to 25 years.

Boeing 777 and 747 Comparison

The introduction of the B777 has posed a dilemma for pavement designers. In this section the B747 is used a reference point for calculating pavement thickness requirements. In this comparison, the 397 tonne version of the B747-400 and the 310 tonne version of the B777 are used. For low strength subgrades, the standard ICAO ACN system rates the B777 as far more damaging than the B747-400. Because the effect for higher strength subgrades is less pronounced, this paper focusses on CBR 3 examples. As shown in the table below, for 10,000 coverages the standard ACN rating implies that the B777 requires 235 mm more pavement depth on a CBR 3 subgrade than does a B747, an increase of about 15%. This would constitute a significant penalty to Boeing in their efforts to market the new aircraft. Yet, because of the different pass-to-coverage ratios of the two aircraft, these figures still understate the real magnitude of the penalty. Ten thousand coverages correspond to 17,500 taxiway passes of the B747 but only 13,400 passes of the B777, and it is aircraft numbers, not coverages that reflect landing fee income. Furthermore, some multiwheeled gears being considered for future generation aircraft would, according to the standard rating system, require up to 65% more pavement thickness than the B747. These predictions were widely thought to be conservative and to result from the rating system’s limitations. This situation is not changed, however, by the introduction of ‘mechanistic’ design methods based on layered elastic analysis. As shown in the table below, both APSDS and LEDFAA predict that the extra pavement thickness required by the B777 is substantial and not significantly different from the 235 mm indicated by the standard ICAO ACN rating system.

A specialist ICAO Study Group has recommended an interim alpha factor of 0.72 for 6-wheel gears, pending resolution of the issue by future full-scale testing. The original deflection-based ESWL design method overstated the damage caused by multiwheel gears. This error was later corrected by introducing the ‘alpha factor’, a pavement thickness reduction factor. Values were derived from full-scale tests in the late 1960s and early 1970s. As shown in the table, the effect of the 0.72 modification is to make the B777 pavement thickness similar to the B747’s at the 10,000 coverage level. Thus, with respect to the treatment of the B777, the Modified ICAO rating system is not consistent with the pavement design methods.

Results:

Model	Thickness (B777)	Thickness (B747)	Difference in Thickness
APSDS 3	2010 mm	1735 mm	275 mm
LEDFAA 1.2	2045 mm	1775 mm	270 mm
ICAO ACN Standard ICAO ACN Modified	1915 mm 1700 mm	1680 mm 1680 mm	235 mm 20 mm

 

RECENT FULL-SCALE PAVEMENT RUTTING TESTS

Boeing recently reported tests conducted for them by Progresstech at Moscow in 1996. The primary purpose of the testing was to measure the rutting damage to flexible pavements caused by the 3-axle (6-wheel) landing gear of the new Boeing 777 relative to dual-axle (4-wheel) landing gears of the kind used on the Boeing 747. The loads per wheel, tyre pressures, wheel and axle spacings were identical for the 4-wheel and 6-wheel test loading rigs. The key unexpected result was that ten thousand passes of each rig produced the same depth of rutting (40 mm). Current pavement design methods assume that the amount of rutting depends primarily upon the maximum compressive strain caused at subgrade level by the vehicle, and upon the number of strain repetitions. In these terms the B777 rig subjected the pavement to 50% more axle loading repetitions than the 4-wheeled rig. Also, computed maximum subgrade strains were significantly higher for the B777 rig. Thus it appears that factors other than those encompassed by current pavement design methods are needed to explain the result. Two concepts appear to be clearly relevant:

Firstly, following the US Army Corps of Engineers’ multiwheel tests in the late 60s and early 70s, Ahlvin (1971) suggested that the better-than-expected observed performance of larger wheel groups might be explained in part by the interior soil confinement provided by the outer wheels of the group. In regard to the 3-axle Moscow rig in particular, restraint provided by the loading from the leading and trailing axles might significantly reduce pavement deformations that would otherwise be caused by the interior axle loading. Secondly, it is widely accepted that repeated loadings are more destructive if the load is totally removed in between applications. Partial removal between applications is observed to produce less damage. It may be the case that a kneading action associated with the increasing and decreasing phases is responsible for the bulk of soil deformation that occurs. The effect of the number of axles may be relatively small. This concept is not incorporated in the current design methods. The limited dropoff in subgrade pressure in between axles that was measured at Moscow may explain in part why similar rutting occurred under the 2 and 3-axle rigs.

CONCLUSIONS

The programs LEDFAA and APSDS, recently introduced into pavement design practice, have many features in common but differ in some respects. LEDFAA is less transparent and, from the point of view of the experienced pavement designer, is less flexible. This inflexibility is necessary, however, because LEDFAA functions as an FAA design standard intended for use in parallel with FAA’s conventional empirical design method. To facilitate the introduction of the new ‘mechanistic’ methodology into design practice LEDFAA was conditioned to produce similar pavement thicknesses to the older method by mandating the properties assigned to materials. During this interim period the mandated values may not reflect the current best assessments of the materials’ properties.

All APSDS inputs, including the performance relationship, can be specified by the user. This flexibility is intended to provide the experienced designer with easy access to the capabilities of a layered elastic based ‘mechanistic’ method. By utilizing available performance data and material properties the user is able to apply customized treatment to individual pavement situations.

The ICAO ACN rating system uses the conventional empirical design method and predicts damage by the B777 and NLAs that appears to be unrealistically high for low strength subgrades relative to that caused by the B747. This situation is not resolved, however, by the introduction of ‘mechanistic’ methods based on layered elastic analysis. Both LEDFAA and APSDS are calibrated against the same limited full-scale tests used to produce the empirical design method and so predict similar B777 pavement damage. Because the pavement design methods remain essentially empirical, confident assessment of the damage by new significantly different aircraft wheel arrangements await the completion of appropriate full-scale tests of the kind programmed for FAA’s National Airport Pavement Test Facility now under construction. Recent rutting tests in which B777 and B747 gears produced equal pavement deformation confirms the need for further full-scale testing.

This does not mean that the introduction of ‘mechanistic’ methods based on layered elastic analysis is not a significant advance in pavement design. The computational power eliminates the need for earlier simplifications such as the single layer theory, the Equivalent Single Wheel Load concept and the Pass-To-Coverage ratio that were necessary to make computations manageable. They also make possible a more rigorous process of calibration against available test or field performance data. They facilitate the introduction of new materials, both bound and unbound, (the task of realistically characterising the stiffnesses of these new materials still remains) and greatly simplify the calculation of the combined effects of mixes of aircraft types and weights. In summary, for aircraft that were represented by the full-scale tests, they provide the designer with a fast tool to design realistic layered pavement structures to cater for complex aircraft traffic mixes.

The examples presented show that, for pavements on CBR 3 subgrades, APSDS and LEDFAA thicknesses are similar. The APSDS results for alternative wander widths show that the pavement thickness can be reduced by about 5% for runway conditions and that the pavement would need to be about 3.5% thicker for a docking bay. Because LEDFAA uses a fixed taxiway wander, it cannot be directly applied to runways and docking bays. Application of taxiway wander to a docking bay designed to have a life of 20 years would reduce the theoretical life to around 12 years.

Postscript (September 2001)

Since this paper was published, we have developed an improved calibration for APSDS.  Click here to view the latest paper.

REFERENCES

Ahlvin, R. G. et al. (1971). Multiple-wheel heavy gear load pavement tests. Technical Report S-71-17, Vol. I, US Army Corps of Engineers, Waterways Experiment Station, Vicksburg, Miss.

Barker, W. and Brabston, W. (1975). Development of a structural design procedure for flexible airport pavements. Report No. S-75-17. US Army Corps of Engineers, Waterways Experiment Station, Vicksburg, Miss.

Barker, W. and Gonzalez, C. (1991). Pavement design by elastic theory. Proc. ASCE Conf. Aircraft/ Pavement Interaction, Kansas City.

Barker, W. and Gonzalez, C. (1992). Design considerations for multi-wheel aircraft. Proc. 22nd Int. Air Transportation Conf., Denver.

Federal Aviation Administration (1995). Advisory Circular 150/5320-16. Airport pavement design for the Boeing 777 airplane. LEDFAA User’s Manual. U.S. Federal Aviation Administration

Federal Aviation Administration (1995). Advisory Circular 150/5320-6D. Airport pavement design and evaluation. U.S. Federal Aviation Administration.

Ho Sang, V.A. (1975). Field survey and analysis of aircraft distribution on airport pavements. Report No. FAA-RD-74-36. U.S. Federal Aviation Administration.

McQueen, R. D., Hayhoe, G., Guo, E., Rice, J. And Lee, X. (1997). A sensitivity study of layered elastic theory for airport pavement design. Proc. ASCE Conf. Aircraft Pavement Technology, Seattle, Washington.

Monismith, C.L., Finn, F.N., Ahlborn, G. and Markevich, N. (1987). A general analytically based approach to the design of asphalt concrete pavements. Proc. 6th. Int. Conf. on the Structural Design of Asphalt Pavements, Ann Arbor.

US Army and Air Force. (1989). Flexible pavement design for airfields. (Elastic Layer Method). Technical Manual TM 5-825-2-1/AFM 88-6.

Wardle, L.J. (1977). CIRCLY Users’ Manual, MINCAD Systems Pty Ltd, Richmond, Vic., Australia.

Wardle, L.J. and Rodway, B. (1995). Development and Application of an Improved Airport Pavement Design Method. ASCE Transportation Congress, San Diego, 22-26 October, 1995.