analysis of existing data · a. report of the development of b-29 design criteria (4).* b. report...

CORPS OF ENGINEERS, U. S. ARMY

DESIGN OF FLEXIBLE AIRFIELD PAVEMENTS FOR

MULTIPLE-WHEEL LANDING GEAR ASSEMBLIES

REPORT NO.2

ANALYSIS OF EXISTING DATA

TECHNICAL MEMORANDUM NO. 3-349

PREPARED FOR

OFFICE OF THE CHIEF OF ENGINEERS

AIRFIELDS BRANCH

ENGINEERING DIVISION

MILITARY CONSTRUCTION

BY

WATERWAYS EXPERIMENT STATION

VICKSBURG, MISSISSIPPI

JUNE U55

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4. TITLE AND SUBTITLE Design of Flexible Airfield Pavements for Multiple-wheel Landing GearAssemblies: Reprot No. 2: Analysis of Existing Data

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CORPS OF ENGINEERS, U. S. ARMY

DESIGN OF FLEXIBLE AIRFIELD PAVEMENTS FOR


REPORT NO.2


TECHNICAL MEMORANDUM NO. 3-349

PREPARED FOR

OFFICE OF THE CHIEF OF ENGINEERS

AIRFIELDS BRANCH

ENGINEERING DIVISION

MILITARY CONSTRUCTION

BY

WATERWAYS EXPERIMENT STATION

VICKSBURG, MISSISSIPPI

ARMY-MRC VICKSBURG. MISS.

JUNE 1955

i

PREFACE

The study reported herein was proposed by the consultants to the

Flexible Pavement Branch, Waterways Experiment Station, in a conference

held on 30-31 March 1953, and was authorized by the Office, Chief of

Engineers, in Addendum No. 5 (fiscal year 1954) dated October 1953 to 11 Instructions and Outline for Multiple-wheel Studies, 11 dated October 1948.

Engineers of the Flexible Pavement Branch who were actively engaged

in directing and carrying out the analysis were Messrs. W. J. Turnbull,

C. R. Foster, and R. G. Ahlvin.

PREFACE

SIDIJMARY . . . . . . . . .

CONTENTS ·

PART I: PURPOSE AND SCOPE OF THE STUDY

Purpose •• Scope •••

PART II: PRESENT TENTATIVE METHOD OF RESOLUTION

PART III: SUMMARY OF PERTINENT DATA

Data from Report on Certain Re~uirements for Flexible

iii

v

1

1 1

3

5

Pavement Design for B-29 Planes • • • • • • • • • • 5 Data from Accelerated Traffic Test at Stockton Airfield

(Stockton Test No. 2) • • • • • • • • • • • • • • • • 5 Design Curves for Very Heavy Multiple-wheel Assemblies~

CBR Symposium, 1950 • • • • • • • • • • • • • • • 6 Data from Investigation of Stress Distribution in a

Homogeneous Clayey Silt Test Section • • • • • • • • • • 6 Theoretical Stresses Induced by Uniform Circular Loads 7 Data from Investigation of Stress Distribution in a

Homogeneous Sand Test Section • • • • • • • • • • • 7 Data from Multiple-wheel Test Section with Lean-clay

Subgrade • • • • • • • • • • • • • • • • 8

PART IV: ANALYSIS

Analysis Based on Deflection • • • • • • • • Analysis Based on Stress Comparison of Design Criteria • Advantage of the Proposed Method of Resolution

PART V: CONCLUSIONS AND RECOMMENDATIONS

. . . . . Conclusions • • • • Recommendations . . . . . . . . . .

REFERENCES

TABLE 1

PLATES 1-22

. . . . .

APPENDIX A: EXAMPLE OF THE COMPUTATION OF EQUIVALENT SINGLE WHEEL LOAD • • • • • • • • • • • • • • • • • • •

PlATES Al-A2

9

9 15 16 17 18 18 18

19

Al-A4

v

SUWA.ARY

This study was conducted for the purpose of re-evaluating the cur

rent, tentatively adopted methods for resolving the existing single

wheel design criteria for flexible airfield pavements into criteria for

multiple-wheel assemblies. Results of tests on the first multiple-wheel

test section indicated that the current method yields design criteria for

pavement and base thicknesses that are slightly on the unconservative

side.

All available data that might provide means of comparing the ef

fects of single and multiple loadings were reviewed and a new analysis

was made. Both stress and deflection effects were examined wherever

possible.

A proposed alternate theoretical means of resolving well

established single-wheel design criteria to give valid multiple-wheel

criteria was developed. This alternate method of resolution is based

solely on equivalent deflections, and appears to give somewhat better

results than the tentative method now in use. The method has the

distinct advantage of being capable of extension to any assembly con

figuration without additional assumptions.

DESIGN OF FLEXIBLE AIRFIElD PAVEMENTS FOR



PART I: PURPOSE AND SCOPE OF THE STUDY

Purpose

1. The purpose of this study was to analyze all available data

pertaining to the relative severity of the effects of single- and

multiple-wheel loadings on flexible airfield pavements, and to deter

mine: (a) whether or not the present tentative method of resolving

single-wheel criteria into criteria for multiple assemblies is ade

~uate; (b) means for obtaining better results if the present method is

not ade~uate; and (c) what additional verification, if any, is needed

for the present method of resolution or for a suggested alternate

method.

2. The study was limited to an analysis of existing data and to

theoretical developments necessary to verify the existing method of re

solving single-wheel criteria into criteria for multiple-wheel assemblies

or to formulate an alternate method. For this purpose, information and

data from the following reports were used:

a. Report of the development of B-29 design criteria (4).* b. Report of the second traffic tests at Stockton Airfield,

California (3). c. CBR Symposium, ASCE (1).

d. Report on stress distribution in a homogeneous clayey silt test section (5).

* Numbers in parentheses refer to the bibliography.

2

/

e. Report on theoretical stresses induced by uniform circular loads (7).

f. Report on stress distribution in a homogeneous sand test section (8).

~· Report of the first multiple-wheel traffic test section (6).

PART II: PRESENT TENTATIVE METHOD OF RESOLUTION

3. The present tentative method of resolving single-wheel cri

teria for design of flexible airfield pavements into design criteria

for multiple-wheel assemblies is explained in detail in the CBR Sym

posium (1). It was first proposed in somewhat less complete form in

the B-29 report (4). This method assumes that at shallow depths each

wheel of a multiple assembly has an individual effect on pavements and

subgrades, while below some greater depth the entire assembly acts as

a single load (see plate 1). Between these depths the wheel-load ef

fects progress in an orderly manner from one to the other.

4. These concepts provide a basis for arriving at a single

wheel load which is considered, for design purposes, to produce effects

on the subgrade equivalent to those produced by a multiple-wheel load.

This equivalent single-wheel load can then be used with the well

validated single-wheel CBR design curves presented in the Engineering

Manual {2) to arrive at designs for a multiple-wheel assembly.

3

5. Depths, above which one wheel of a multiple assembly is con

sidered to act as a single load and below which the entire assembly can

be considered as a larger single load, have been established from theo

retical and empirical data (1), (4). These depths have been expressed

in terms of dimensions of the assembly configuration. The sketch shown

below will help clarify an explanation of these dimensions.

Dual Dual-tandem

0 The least distance between adjacent contact areas is designated as "d."

The depth above which one wheel of an assembly is considered as a single

wheel has been empirically established as one-half of this distance, or d 2. The greatest distance (center to center) between any two wheels of

an assembly is designated as "S." The depth below which the assembly

load is considered to be a single load has been empirically established

4

as twice this distance, or 11 28." Between these two extr~:nes, equivalent

single-wheel loads are determined from a straight-line relation on a

log-log plot. Plate 2 includes examples of this type of development

for 150,000-lb assembly loads. In the case of the dual assembly, for

instance, the load on one wheel of the assembly (75,000 lb) is the

critical load for base and pavement thicknesses less than 12.6 in. The

entire assembly load (considered as a single-wheel load) is critical

for thicknesses greater than 112 in. Between these two points (75,000

lb at 12.6 in. and 150,000 lb at 112 in.) equivalent single-wheel loads

are represented by the straight line shown on plate 2.

5

PART III: SUMMARY OF PERTINENT DATA

6. At the time the present tentative method of resolving single

wheel design criteria into criteria for multiple-wheel assemblies was

formulated (August 1945), only a very limited amount of pertinent data

was available. Since that time a number of investigations have produced

directly comparable data for single- and multiple-type loadings, i.e.,

stress and deflection data which can be compared for single and multiple

loadings. In one instance, that of the multiple-wheel test section

tested in 1949 and 1950 at the Waterways Experiment Station, simulated

aircraft traffic was applied to the test section with multiple-wheel

gear. The results of these accelerated traffic tests indicated that

the design curves developed using the present method of resolution give

values which are slightly on the unconservative side. The data which

provided a basis for the current study are described in the following

paragraphs.

Data from Report on Certain Requirements for Flexible Pavement Design for B-29 Planes (4)

7. The B-29 report, in addition to developing the present tenta

tive method of determining criteria for multiple-wheel landing gear

assemblies, presents the results of "Flexible Pavement Tests, Marietta,

Georgia." The Marietta tests provided stress and deflection measure

ments beneath B-24 single and B-29 dual wheels for a range of loads on

four thicknesses of pavement and base overlying a weak subgrade. Data

from tables 3 and 10 of the B-29 report were used in the current anal

ysis, and plates 3 and 4 are taken directly from the B-29 report.

Data from Accelerated Traffic Test at Stockton Airfield (Stockton Test No. 2) (3)

8. The second Stockton test section included 18 items of various

thicknesses divided generally into strong, medium, and weak subgrade

6

groups. Comparable single- and multiple-wheel stress and deflection

data are available for all these items. While the testing at Stockton

No. 2 included accelerated traffic tests using single loads, no traffic

with multiple assemblies was applied; therefore, no equivalent results

with traffic are reported. However, rather complete data comparing the

effects on stresses and deflections produced by standing loads of single

and multiple assemblies are included. The data pertinent to the anal

ysis reported herein are shown in the Stockton report as exhibits I-9

through I-12 of appendix D, exhibits I-2 through I-19 of appendix c, and

exhibits I-40 through I-44 of appendix F. Plots of vertical stress

versus depth based on these data are shown on plate 5.

Design Curves for Very Heavy Multiple-wheel Assemblies, CBR Symposium, 1950 (1)

9. This article includes the data presented in the B-29 report

and is included here because it extends the multiple-wheel developments

a little further than does the presentation in that report. Plates 1

and 2 which are taken from the CBR Symposium portray the concepts that

formed the basis of this method of resolution and illustrate the method.

Data from Investigation of Stress Distribution in a Homogeneous Clayey Silt Test Section (5)

10. The Waterways Experiment Station is currently studying the

distribution of stresses and deflections in soil masses in connection

with a long-range investigation aimed at the development of more ra

tional methods of design of flexible pavements. Thus far, two homo

geneous test sections have been constructed and testing thereon com

pleted. The results of tests on the first of these, the clayey silt

test section, include stresses and deflections measured in a homogene

ous soil mass beneath static, single and dual, uniform circular loads.

The single loads were of 1000-sq-in. circular area, while the dual

loads consisted of two 500-sq-in. circular areas. Load intensities

of 15, 30, 45, and 60 psi were applied through the loading plates, and

measurements were made at various offsets and depths such that stress

and deflection versus offset curves could be developed for 1-, 2-, 3-,

4-, and 5-ft depths. Dual spacings of 3, 4.5, 6, and 7.5 ft were used.

Deflection data used in the current analysis were extracted from plates

103 through 107 of the clayey-silt test section report and are shown

on plate 6 herein. Plates 7 and 8 herein, which were taken directly

from that report, are typical of normal and shear stress data presented

in the report.

Theoretical Stresses Induced by Uniform Circular Loads (7)

11. The report on theoretical developments in connection with

7

the stress distribution studies presents formulas developed from work

done by A. E. H. Love (9) in 1929 that can be used to compute the

stresses and deflections in a semi-infinite, homogeneous, elastic mass

subjected to a uniform circular load. Stresses and deflections computed

from these formulas are used in this analysis. The theoretical deflec

tions on plate 9 were computed in this manner.

Data from Investigation of Stress Distribution in a Homogeneous Sand Test Section (8)

12. The results of tests on the hcmogeneous sand test section

have recently been published. They include stresses and deflections

measured in a homogeneous mass of dry sand beneath static, single and

dual, uniform circular loads. Three plate sizes, 250, 500, and 1000

sq in., were used for both single and dual loadings. Intensities of

15, 30, and 60 psi were applied and dual spacings were included as

shown in the following table.

Plate Size, sq in.

250 500 500

1000

Dual Spacing, ft

2.5 3.0 6.0 4.5

8

Measurements were made at sufficient points in the mass so that stress

and deflection versus offset curves could be developed for 0.5-, 1-,

2-, 3-, 4-, and 5-ft depths. Pertinent data were used in preparing

plates 10 and 11. Plates 12 and 13 were taken directly from the report

on the sand test section; maximum deflections from these measurements,

which were used in the multiple-wheel analysis, are shown in table 1.

Data from Multiple-wheel Test Section with Lean-clay Subgrade (6)

13. A test section was constructed and tested at the Waterways

Experiment Station to establish the validity of the present tentative

method of developing design criteria for multiple-wheel landing gear

assemblies on flexible airfield pavements. This test section was built

of crushed limestone on a processed lean-clay subgrade and was surfaced

with a 3-in. layer of asphaltic concrete. Two parallel lanes were in

cluded, each divided into three parts. The thickness of the central

section of one lane was determined from the tentatively established

criteria for a B-29 assembly loading. The end sections were, respec

tively, a 30 per cent underdesign and a 30 per cent overdesign in

terms of thickness. The second lane included, similarly, a 30 per cent

underdesign, correct design, and a 30 per cent overdesign for a B-36

assembly. These lanes were subjected first to traffic with the as

semblies for which they were designed and subse~uently to heavier

loadings.

14. It was concluded from this study that the present tentative

method of deriving multiple-wheel design curves gives criteria slightly

on the unconservative side. Results of the study that are pertinent to

this analysis are included as plates 14 and 15 which are modifications

of plates in the "multiple-wheel report."

9

PART IV: ANALYSIS

15. The present method of resolution of single- into multiple

wheel design criteria was based on a compromise between stresses and de

flections used as a basis for arriving at a single-wheel load equivalent

for design purposes to the multiple-wheel load. By this method depths

were determined above which individual wheels act independently and below

which multiple-wheel assemblies act as a unit. Between these two points

a straight-line relation on a log-log plot was accepted as a simple yet

satisfactory representation of the variation in effective single-wheel

load. The reanalysis is made on the basis of both deflection and stress

using the theory of elasticity to compute equivalent single-wheel loads

rather than the two established points and an approximate geometric rela

tion for intermediate points. Curves developed on the basis of deflec

tion and stress are compared with available traffic behavior data.

Analysis Based on Deflection

Original analysis

16. The original analysis of deflection data considered that strain

was an important criterion and that the critical strain is represented by

the rate of change of deflection with offset along the deflection pro

file. It also considered that the effects produced by a dual loading

could be produced by a single load with the same pressure intensity having

a gross magnitude between that of one and both wheels of the dual. These

considerations are reasonable and remain unquestioned by any reanalysis.

17. Although the slope of deflection profiles was accepted as an

important criterion, data were not adequate to develop such profiles at

the time of the original analysis. It was therefore assumed that the

maximum deflection was representative of the critical slope and that the

maximum deflection for a dual assembly occurred beneath the center of

one wheel of the assembly. With the additional data now available, de

flection profiles can be developed and the magnitudes and positions of

maximum deflections beneath multiple-wheel assemblies can be reasonably

determined.

10

18. The crucial element of the original development as regards

deflection data is shown in the plots on plate 3. Deflection-depth rela

tions were empirically shown to be straight-line r~lations on log-log

plots. The lines representing these relations for a dual assembly, one

wheel of the assembly, and a single wheel of the same gross load as the

assembly, were then plotted on a single log-log plot (see plate 3). These lines were extended well beyond the range of available data such

that the dual-load line intersected each of the other lines. The inter

section with the smaller single-wheel-load line gave a depth above which

the dual load was considered to have the same effect as the single,

while the intersection with the larger single-wheel-load line gave a

depth below which the dual assembly was considered to have the same ef

fect as the larger single load.

19. Both theoretical data and later test data show that the rela

tion between thickness and deflection is not well represented by a

straight line on a log-log plot. Plate 10, which was developed from

later data, shows this quite clearly. Straight lines can be used to

represent the true relation for narrow ranges in depth, and this was

done for the analysis discussed in the previous paragraph, but even

small extrapolations of these straight lines gave undesirable errors.

Both theoretical and later test data also showed that for commonly used

spacings the second wheel of a dual assembly contributes an appreciable

portion of the maximum deflection even for depths near the surface. This

is shown by the single-wheel curves for deflect:~on versus offset on

plates 6, 9, and 11 where it can be seen that appreciable deflections

are produced by a single wheel at offsets commensurate with reasonable

dual spacings. One wheel cannot, therefore, be considered to act inde

pendently even at very shallow depths. The over-all effect of these

discrepancies is not large and the design curves developed by the cur

rent methods of resolution are only slightly unconservative.

Multiple-wheel test section

20. The results of traffic-testing with multiple-wheel assemblies

are indicated on plate 14. The first "multiple-wheel report" also in

cluded an analysis based on deflection, the results of which are shown

11

on plate 15. Both of the analyses in the multiple-wheel test section

report tend to show that the design criteria in present use give designs

that are slightly unconservative.

New method of resolution

21. Since the present tentative method of resolving single-wheel

criteria to criteria for multiple-wheel assemblies appeared to be some

what inadequate, development of an alternate, better method was consid

ered desirable. The reanalysis of stress data described later in para

graphs 35 through 38, indicated no variance with the earlier concepts on

which the present method was based. However, the reanalysis of deflec

tion data including analysis of more recent data showed that better

limiting assumptions can now be made which will give somewhat more

realistic results. A new method has therefore been developed which is

considered to give criteria consistent with both stress and deflection

data and to be in better agreement with traffic test results.

22. Failure is produced in a pavement system by a movement or dis

location of material. This movement is manifested as strain or deflec

tion. It is reasonable, therefore, to look to strain or deflection as a

criterion of failure. Little or no strain data are available, but as was

pointed out in the original analysis and is re-emphasized in this anal

ysis, it is reasonable to accept the slope (rate of change) of a deflec

tion versus offset curve as indicative of the critical strain.

23. Thus, if it can be shown that a multiple-wheel load which

produces a maximum deflection equal to that of a single-wheel load yields

deflection versus offset curves at various depths whose slopes are less

than those for the single load at equal depths, it may be concluded that

the multiple-wheel assembly is creating no more severe strains than the

single wheel. Pertinent data are available from the stress distribution

studies (5), (7), (8). Plate 9 (theoretical developments) and plates 6 and 11 (results of tests on the homogeneous clayey-silt and sand test

sections) show the relation between deflection versus offset curves at

equal depths for single and dual assemblies. Without exception the

slopes of the deflection versus offset curves for the single loads are

equal to or steeper than those for the dual loads at equal depths.

12

24. From this analysis it appears that a single-wheel load which

yields the same maximum deflection as a multiple-wheel load will produce

equal or more severe strains in the subgrade or base than will the

multiple-wheel load. The single load may, therefore, be considered

equivalent to the multiple-wheel load for purposes of design, and it is

proposed that the existing well-validated single-wheel curves and this

equivalent single-wheel load be used to develop designs for multiple

wheel assemblies.

25. The slopes of some of the single-wheel deflection profiles in

plates 6, 9, and 11 are appreciably greater than their dual-wheel counter

parts. Therefore, for design purposes, it might be considered that as

suming the single-wheel loads equivalent to their dual counterparts

would introduce too much conservatism. As will be shown later, however,

the proposed method gives design criteria only a little more conserva

tive than that currently used, which has been shown to be slightly on

the unconservative side.

26. Comparison of the theoretical curves of plate 9 with the test

data curves of plates 6 and 11 shows that the theoretical curves are

similar in general form to those derived from test data, and that for

all but the shallowest depths the similarity of the curves is quite

close. At the shallow depths discrepancies occur for the wide offsets,

but at these depths the maximum deflections for a multiple assembly are

almost entirely the result of the load on one wheel. For this reason

discrepancies at wide offsets can have only a slight effect, and it is

therefore considered that theoretical deflections can be used in ar

riving at the relation between single- and multiple-wheel assembly loads.

Determination of equivalent single-wheel load

27. Each wheel of a multiple-wheel assembly contributes a part of

the maximum deflection occurring at any depth beneath such an assembly.

Curves of deflection versus offset for various depths for a single wheel

can be determined theoretically. Reference (5) includes a set of theo

retical curves from which deflections at any offset and depth can be in

terpolated. Reference (7) describes methods and gives formulas that

13

permit the direct computation of deflections at any offset and depth.

From the single-wheel curves, curves of deflection versus offset can be

developed for multiple-wheel assemblies by use of the principle of super

position. Using single- and multiple-wheel curves of this type the maxi

mum deflections at a given depth beneath single- and multiple-wheel as

semblies can be determined. By e~uating these deflections a relation

between multiple-wheel and equivalent single-wheel loads can be estab

lished. In equating these deflections the contact area of the single

wheel is taken to be constant and the same as that of one wheel of the

multiple assembly. By determination of the equivalent single-wheel load

for a number of depths throughout the pertinent depth range, a relation

between depth and e~uivalent single-wheel load can be established. This

relation can then be used to resolve the established single-wheel design

criteria into criteria for multiple assemblies. An example of the de

termination of equivalent single-wheel load is given in appendix A.

28. Design curves developed by use of this method were produced

for capacity operation and are shown by dotted lines on plates 14

through 22, inclusive, for the various assemblies represented in these

figures. For comparison, the corresponding curves, based on current cri

teria,are shown on these plates by solid lines.

Validation of new method of resolution

29. Analyses similar to those used in processing the data from the

multiple-wheel test section (6) have been made wherever data were ade~uate.

These consist of developing curves of single-wheel load versus maximum

deflection from test data then using these curves with the maximum deflec

tions occurring beneath multiple-wheel loads to determine an equivalent

single-wheel load. By using this e~uivalent single-wheel load,single

wheel CBR design curves can be used to arrive at the required thickness

for the multiple-wheel load.

30. Multiple-wheel test section. Design curves developed based on

the proposed method are plotted on plates 14 and 15 along with the present

design curves. In addition to these curves plate 14 shows points in

dicating test section behavior under traffic of 2000 coverages. Plate 15,

14

in addition to the curves, shows points that represent design thicknesses

based on equivalent single-wheel loads determined as stated in the

previous paragraph. Plate 14 shows the new curves to be in better agree

ment with the plotted points than are the curves developed using the

present method of resolution. On plate 15 only the 0 coverage points

are completely valid and these show the new curves to be a better cor

relation than the old. The points for larger numbers of coverages are

not based on completely comparable deflection data since single-wheel

deflections were measured only at 0 coverages.

31. Marietta test section. No traffic data were collected during

the Marietta tests, but sufficient data on deflection under standing

loads, both single and dual, are available for an analysis based on

equivalent single-wheel loads. Such analysis was made in the same way

as for the multiple-wheel test section data. The results are presented

on plate 16 together with design curves determined from both the present

and proposed methods of developing criteria.

32. Stockton test section No. 2. No traffic data for multiple

wheel assemblies were collected during the Stockton tests, bu~ a con

siderable amount of deflection data was assembled. Accordingly, an anal

ysis was made based on equivalent single-wheel load as was done for the

Marietta data. Results are presented on plate 17 along with present and

proposed design curves. Here, again, as with the multiple-wheel test

section (paragraph 30), some of the equivalent deflection data (single

multiple) are not for comparable numbers of coverages. Only 0 coverage

multiple-wheel deflections were measured and 0 coverage single-wheel de

flections were not reported in every case.

33. Stress distribution test sections. Deflection data are avail

able from both the clayey silt and sand test sections used in the stress

distribution investigation. These data have also been analyzed on the

basis of equivalent single-wheel loads as was done with the Stockton and

Marietta data. Results are shown on plates 18 through 21. Since no

data for a single load of the same contact area as one of the duals were

available for the clayey silt, a conversion of the available single-load

data was made, based on theoretical concepts.

15

34. Results. The service behavior analysis (plate 14) shows bet

ter agreement between the plotted points and the design curves based on

the proposed method of resolution of single- to multiple-wheel criteria

than between the points and curves based on the present method. This

provides the strongest validation of the proposed method. The various

other analyses tend to support this validation. For these latter, it

might be argued that the criteria and validation analyses have something

of the same basis. On the other hand, no better means of analysis was

found and without it the bulk of available data could not be compared.

Analysis Based on Stress

Original analysis

35. The analysis of theoretical stress data as presented in the

B-29 report (4) remains unchanged and is quite valid. Plate 4 presents

this analysis together with vertical stress measurements from the

Marietta tests (4). It indicates that, at depths less than about 16 to

20 in., the effect of a B-29 dual assembly on the subgrade is the same

as or less severe than the effect of a single wheel equivalent to one

wheel of the dual. The analysis further indicates that the B-29 dual as

sembly produces much the same amount of stress below a depth of about 75

to 80 in. as does a single wheel having the same gross load. Since these

relations are largely theoretical, the depths can be considered in terms

of radii of the loaded area to give evidence in substantiation of the

d/2 and 2S method of resolution (see paragraph 5).

Later pertinent data

36. Additional evidence has become available that shows the dis

tribution of stresses beneath wheel loads or simulated wheel loads to be

much as indicated by computations based on the theory of elasticity

(Boussinesq theory). Plates 5, 7, 8, 12, and 13 present test results

that bear out this conclusion. Plate 5 was prepared from the Stockton

No. 2 test data and from computations using the methods presented in the

stress distribution report on theoretical developments (7). Plates 7

and 8 were taken directly from the clayey silt test section report (5).

16

Plates 12 and 13 have been prepared for use in the report on the sand

test section studies (8).

Reanalysis

37. The data just presented serve to prove that it is valid to

use theoretical stresses in lieu of actual stresses for analysis. Such

theoretical stresses formed most of the basis for the original analysis

insofar as it was based on stresses. Since this reanalysis only strength

ens the original analysis and since the original analysis led to inade

quate results, the correlations serve to further prove the inadequacy of

normal and shearing stress as a basis for relating the effects of single

and multiple loadings.

Equivalent single-wheel load based on maximum shear stress

38. In the current studies, maximum shear stresses were also tried

as a basis for developing multiple-wheel design criteria. By using the

procedures outlined in paragraph 27 a method of resolving single-wheel

criteria into multiple-wheel criteria, similar to the proposed method

but using shear stress instead of deflection, was evolved. From this

design curves for the B-29 were developed. In general, these curves are

even less conservative than those using the present method of resolution

(see plate 22). Therefore the method is not considered worthy of further

pursuance.

Comparison of Design Criteria

39. In order to show the effect on designs of the use of the pres

ent, proposed, and shear-stress methods of resolution, curves on both

semilog and log-log plots have been developed for the B-29 airplane.

These are shown on plate 22. Comparison reveals that the proposed method

for developing multiple-wheel design criteria gives results roughly

equivalent to, yet somewhat more conservative than, the present method.

The shear-stress method yields the least conservative criteria of the

three.

17

Advantage of the Proposed Method of Resolution

40. If the relation between single and dual computed deflections

is accepted as adequately representing that for actual deflections, the

proposed method provides a rational relation between multiple- and

equivalent single-wheel loads. Thus, any new configuration of landing

gear wheels can be handled as readily as those now existing.

18

PART V: CONCLUSIONS AND RECOMMENDATIONS

Conclusions

41. Based on the analysis herein pertaining to the development of

criteria for designing flexible airfield pavements for multiple-wheel

landing gear assemblies, the following conclusions appear warranted:

a. The present tentative method of resolving single-wheel into miltiple-wheel designs gives criteria slightly on the unconservative side.

b. Neither vertical stress nor maximum shear stress provides an adequate basis for relating the effects of single-and multiple-wheel assemblies.

c. Strains, which are in effect the slopes of deflection versus offset curves, provide the best basis for arriving at single-wheel loads that are equivalent, for design purposes, to multiple-wheel loads.

d. These strains are adequately represented in relative magnitude by theoretical maximum deflections, and satisfactory design criteria for multiple-wheel assemblies can be developed from established single-wheel criteria on the basis of equal maximum deflections.

Recommendations

42. Based on knowledge gained from the analysis or reanalysis of

available data pertinent to design of flexible airfield pavements for

multiple-wheel assemblies, the following actions are recommended:

a. Adoption of the method of resolution proposed herein as the basis for developing design curves for multiple-wheel landing gear assemblies from well-validated single-wheel design curves.

b. Construction and testing of a second test section involving greater design thicknesses to provide data for checking the validity of design criteria for weaker subgrades. In this respect it appears that unusual wheel configurations need not be used. It is also recommended that only a surface treatment be used on the test section in order to eliminate the effects of temperature, and perhaps other factors, on pavement strength.

19

REFERENCES

1. Boyd, H. K., and Foster, C. R., "Design curves for very heavy multiple wheel assemblies, development of CBR flexible pavement design methods for airfields, a symposium." ASCE Transactions, vol 115, p 534 (1950).

2. Corps of Engineers, Office, Chief of Engineers, Engineering Manual for Military Construction. Part XII, chapter 2, July 1951.

3. Corps of Engineers, Sacramento District, Accelerated Traffic Test at Stockton Airfield, Stockton, California (Stockton Test No.2). Sacramento, California, May 1948.

4. Corps of Engineers, Waterways Experiment Station, Certain Requirements for Flexible Pavement Design for B-29 Planes. Waterways Experiment Station, Vicksburg, Miss., August 1945.

5. , Homogeneous Clayey-silt Test Section, Report No. 1, Investigations of Pressures and Deflections for Flexible Pavements. Waterways Experiment Station, Technical Memorandum No. 3-323, Vicksburg, Miss., March 1951.

6. , Test Section with lean Clay Subgrade, Report No. 1, Design of Flexible Airfield Pavements for Multiple-wheel Landing Gear Assemblies. Waterways Experiment Station, Technical Memorandum No. 3-349, Vicksburg, Miss., September 1952.

7. , Theoretical Stresses Induced by Uniform Circular Loads, Report No. 3, Investigations of Pressures and Deflections for Flexible Pavements. Waterways Experiment Station, Technical Memorandum No. 3-323, Vicksburg, Miss., September 1953.

8. , Homogeneous Sand Test Section, Report No. 4, Investiga-tions of Pressures and Deflections for Flexible Pavements. Waterways Experiment Station, Technical Memorandum No. 3-323, Vicksburg, Miss., December 1954.

9. Love, A. E. H., "The stress produced in a semi-infinite solid by pressure on part of the boundary." Philosophical Transactions of the Royal Society, Series A, vol 228, pp 377-420.

Table l

Maximum Deflections Hcmogeneous Sand Test Section

Surface Plate Load Maximum Deflection in Inches for Depth Size Spacing Intensity 0.5 1 2 3 4 5

sq in. ft psi ft ft ft ft ft ft -- --250 Single 15 .0165 .0081 .0024 .0019 .0009 .0015

30 .0210 .0126 .0048 .0030 .0012 .0048 6o .0420 .0276 .0120 .0060 .0036 .0024

2.5 15 .0225 .0096 .0053 .0021 .0023 .0015 30 .0240 .0168 .0072 .0042 .0045 .0018 6o .0480 .0312 .0144 .oo84 .oo48 .0036

500 Single 15 .0183 .0123 .005.1_ .0030 .C024 .0024 30 .0252 .0186 .0102 .0063 .0030 .0018 60 .0516 .0420 .0216 .0120 .oo6o .0036

3.0 15 .0318 .0176 .0069 .0039 .0030 .0024 30 .0324 .0216 .0132 .0084 .0048 .0036 60 .0600 .o468 .0276 .0168 .0108 .0096

1000 Single 15 .0263 .0180 .0114 .0050 .0038 .0024 30 .0357 .0268 .0150 .0099 .oo6o .0054 60 .0751 .0595 .0320 .0210 .0144 .0096

4.5 15 .0330 .0228 .0081 .0069 .0051 .0033 30 .0390 .0324 .0192 .0108 .0102 .oo66 60 .0816 .0720 .0456 .0300 .0240 .0204

~

I> p

·~~~?· p

SUBGRADE . . - I " SHALLOW BASE

fiG A

PLATE TAKEN DIRECTLY FROM CBR SYMPOSIUM

A

"' ,

"

!• s --------1

•

BASE

&

SUBGRADE DEEP BASE

fiG B

WEARING COURSE

.. ..

4

t>

SCHEMATIC DIAGRAM OF B - 29 DUAL WHEEL ASSEMBL 'V

\J r ~ 1"1

N

200,000

150,000

miOO,OOO ..J I

0 <( 0 ..J 75,000 ..J w w ~ 60,000

50,000

37,000

30,000

1 I I I I I T I I I I I I

-CALIFORNIA BEARING RATIO " " CASE I SINGLE--.._ 15 10 9 8 1 6 5 4 3 ./2S=I12,/25=146

II ~~~JI II 1 ~, I ~ ~ ~ ~~ I v

1 II;~ ~ ~ v / I Ill I I ~ / l I~ 1- I

/ I-f- c~ SE m DUAL trANDEM

CASE II DU I'LS, I/'~ 1;1 11 ~' II' / ~ """"

f-12.6'1 _>t ~ II I /7 i' I y I

~ / (((J v(f v / I 1/R rJ I I I I/ VIII/ 71 I

~ v ~ Vff/J /; II I/ ....,__ D" 7 1/; rill) VI 8 9 10 15 20 30 40 50 60 70 80 100 150 200

THICKNESS OF BASE AND PAVEMENT -INCHES

NOTE: TAI<EN DIRECTLY FROM CBR SYMPOSIUM. TENTATIVE METHOD OF COMPARING

THICKNESS REQUIREMENTS FOR VARIOUS WHEEL LOAD ASSEMBLIES

0824530

MARIETTA

DEF'LECTION -THICKNESS

_lj

1\ MOVING _ll L1

\\ \

\\ \ \

'~ 0 .o 10 ...____._......._.......__ .......... _._._._ ..................... 11~ 10 20 30 40 50 80

THICKNESS- INCHES

FIGURE A

TAKEN DIRECTLY FROM REPORT ON FLEXIBLE PAVEMENT DESIGN FOR B-29 PLANES

0901$~ A

(/) w :r 1.) z

z 0 j: 1.) w ..J u. w 0

w 0 < a: ~ al

1.0 9 8 7 6 5

4

3

2

::::> 0.10 (/) 9

8 7 6 5

4

3

2

0.010

MARIETTA

DEFLECTION -THICKNESS

_l _l_l j_l

I~ ll \\

STANDING \\

\\ \\ \\ \j\

1\\ i\ 6( -~ ~~ I!~G I/ L

\ \ \ \

\ \ 0 lfiP 0( L ~.

\ \ If' .lX -'-' 1\ ;\

l1 ' ·, 1\ i\1\

\

ll 30- IP 'II\ IGL £1\ 1'.

IU.L 1"":1

_l

~·. j

r\

ll 10 20 30 40 50 80

THICKNESS -INCHES

FIGURE B

SUBGRADE DEFLECTIONS SINGLE VS DUAL WHEELS

PLATE 3

., r ~ rn ~

0

10

20

30

., ... :z: v z -;-40 :z: 1-L ... 0

50

10

10

10 0

DEPTH- MAXIMUM VERTICAL STRESS

....,.. ~ ·-,. ~ iuAL WHI. ~L-> I--f--~ / .J4CI po-L , s 'VGL~ ,.., I£EL I--~

- ... ---f...-'""" ~ _....

~ "' r- .....; ,...... ,.,,.. ESSJ S .~ / c T~D V£R ICA ST

b? ~ / 06 / -

// k 60. DO- • Sl WGL WM £L

/~ '

I 'fi l I / I

i/of i f, PftE$$URE CELL REAOINC.$

I ., 0 II- 24 PLANE ll 8- 2t PLANE

r ., PLOTT£0 POINTS ARE F'ROII TABLES 7 AND 12,

I I MARIETTA REF'Of\T OATEO MAY, 1145

~ ~

10 20 30 40 so eo 10 10 IIA XI MUM VERTICAL STRESS LBS/ SQ IN

FIGURE A

TYPE LOAO OF' PER TIRE MAJOR

WHEEL LB AXIS-IN.

SINGLE (11-24) lQOOO 27.12 DUAL C8-2t) 30,000 27.12 SINGLE SQOOO 311.35

ELUPTICAL CONTACT AREAS CONCENTRATION FACTOR CNl -3

NOTE:

MINOR AXIS-IN.

1$.73 15.73 22.24

TOTAL CONTACT

AREA SQ IN.

335 870 870

CONTACT OUALS PRESSURE c-c LII/SQIN. IN.

1111.55 &!1.55 37.0 411.55

I. COMPUTATIONS OF' STRESSES WERE MAOE USING NEWMARI<'S STRESS CHARTS ANO OTHER GENERALLY ACCEPTED STAESS FORMULAE.

0

10

20

30

., w :z: v z ;"40 :z: IlL

~ so

10

110

~OM

""()

0

Pun: !> SH.

po-L ,.., ~/' EL

;, I I I r;

I

I I

II

DEPTH-MAXIMUM SHEAR STRESS

·-~ I D AL "'H£ L-~

~R ST"R SSE ~ ~ :;;...-

/ ~ /.

VGL£ r [,/ t--... A

/, 7 //

lj / '14 poo-L It $1 'IGL£ WHl /

'

I

20 25 10 MAXIMUM SHEAR STRESS Lll/ SQ IN

FIGURE B

~

~v ~/

L

30

"fAt< EN OIREC"fLY FROM REPOR"f ON

FLEXIBLE PAVEMENT DESIGN FOR B-29 PLANES

VERTICAL AND SHEAR STRESSES

I

2

3

0 <(4 a: I

X le;5 0

6

7

8

9

I

2

3

0 ~4 I

X 1-:!;5 0

6

7

8

9

20 40 Oi -PSI

eo eo 100 120 140

v v

/ Vo

I 00

ofe

Ia>

WHEEL LOAD 36,000 LB

CJ'z -PSI 20 40 60 80 100 120 140

............ k. r OOO:l

op XIXJ)(I)

/'0 1'frm

I I


LEGEND --- THEORY (BOUSSINESQ)

o POINTS FROM STOCKTON TEST 2

082.7.53-H

I

2

3

0 ~4 I

X 1-:!;5 0

6

7

8

9

I

2

3

0 <(4 a: I

X .... 1!;5 0

8

7

8

9

C1'z-PSI 20 40 so 80 100 120 140

~ v

/ ~ 0

.~ )

f I


CJ'z -PSI 20 40 60 80 100 120 140

~V /

~ 00

Vc 0

ko I

1 J


STRESS VS DEPiH <iz AT 0 OFFSET

SINGLE LOADS

PLATE 5

'1J r ?:j IT! 0)

0827!l3C

0 0.00

0.02

..,o.o• ... ::z: ~ 0.08

~ 0.08

~ t 0.10 .., .J ::; 0.12 Q

0.1.

0.18

0.18

~

~

'

OFFSET IN FEET 2 3 4 s 8 1 8 v 10 11

DUAL l--2 L

If " / : v r

I 17 r-t 1StN~fE

1.0-FOOT DEPTH

OFFSET IN FEET 2 3 • s 8 1 a 9 10 11 ::l o.oo 0

::z: f--o.-( DUAL u ......... ~ .--!o.o2 ~

~ 0.0. t= :.l 0.08 .J ... "' Q 0.08

I :-t--

./ ........ ~/ v

~ v

-{1StN~LE

3.0-FOOT DEPTH

LEe; END

DUAL LOAD DEFLECTIONS SINGLE LOAD DEFLECTIONS

0 0.00

OFFSET IN FEET 2 3 4 S 8 7 8 9 10 II

r- t DUAL ~ ~·-.....

0.02

0.04 ., "' ~ o.o8 ~ ~ 0.08

~ i= 0.10 u "' ~ 0.12 w Q

0.14

0.18

0.18

I v If

_II J~

N , ~.__, SINGLE

2.0-FOOT DEPTH

NOTE' SOO-SQ-IN. PLATE, 3.0-FT DUAL SPACING.

SINGLE LOAD DEFLECTIONS WERE INCREASED &Y RATIO TO MAKE MAXIMUM DEFLECTIONS FOR SINGLE AND DUAL LOADINGS EQUAL.

SOO-SCHN. SINGLE LOAD DEFLECTIONS WERE OBTAINED AT HOMOLOGOUS POINTS FROM IOOO..SQ-IN. SINGLE LOAD DATA.

DEFLECTIONS WERE AVERAGED FROM THOSE PRODUCED &Y IS-,30-;45-AND 50-PSI SURFACE LOAD INTENSITIES.

COMPARISON OF SINGLE AND DUAL DEFLECTION PROFILES

TEST DATA

CLAYEY-SILT TEST SECTION

I SINGLE J 100

90

80

70

60

50

40

30

20

10

0

"' ~ 100 U) U)

~ 90 a.

.... 80 v ;<: z 70 0 v w 60 v ~ g; 50 U)

~ 40

... z 30 w v a: w (L

I U) <I)

"' a:

20

IO

0

I

I

~

'\

~ l I

I I

\I ~ \\ ~

1"\. 2

--I

h---<~ 1 ~\ In ~~

J,j \ {! \

4 5 6 10 OFFSET - FEET

---RIGHT DUAL 3.0-FT SPACING

I I I I :

~ ... <I) 0 3 4 ~ 6 10

01101538

100

90

80

70

60

50

40

30

20

J 10

~ 0

0

I

/· II/ i1 i/

...__\

f"\,

OFFSET - FEET

-· ------ RIGHT DUAL

I 4.5-FT SPACING

l

l \ ~

4 5 6 OFFSET - FEET

IO

~

u ;<:

90

80

70

60

50

40

30 r--20 f--

IO

z 0 u

~ 0 0

... 0

... ~ 100 u

ffi 90 n. I

<I) 80 <I) w a: ... 70 .,

60

50

40

30

20

10

I

0 0

RIGHT DUAL 6.0- FT SPACING

,.. // \\ /i/ ~\ ~I \

~I

1-/ I\\ ~;

~~ l1 '~ ~

4 5 6 8 OffSET -FEET

- .1.7$'-··-RICHT DUAL ,

I I 7.5-FT SPACING

~

' : j I I I

1/ ,\ I

/~ ~\I v \\

\\ I \ I

I \ \ ~

4 5 6 OFFSET - FEET

LEGEND X 15,000-LB LOAD 0 30,000-LB LOAD l>. 45,000-LB LOAD C 60,000-LB LOAD • ALL LOADS

--THEORETICAL, N•3 -----THEORETICAL N•4 ---THEORETICAL: N•5 POISSON'S RATIO • 0.5

NOTE: OFfSET MEASURED FROM CENTROID OF LOADED AREA ALONG )(-AXIS

8

IO

IO

TAKEN DIRECTLY FROM REPORT ON HOMOGENEOUS CLAYEY- SILT TEST SECTION

STRESS VS OFFSET DISTANCE <J; AT 1-FT DEPTH

PLATE 7

I SINGLE l 0

---~~ ...... r-1 PSI ./ I

-["20P$1 \: N 2 ~

~- / \ .......

"' 3 t.,ops/.~ _/

.... k-./ ...--lol ..

:::.-- 'LsPSI '--......_ lol .. I ~ l: ....

I ... e I!! I

1

a

e

10 0 2 3 4 3 5 8 10

OFFSET - FEET

~I.S'1 RIGHT DUAL

10 PSI I 3.0 FT SPACING

O ~~l,i$PtJ ~F=._' I~~~)) ~

2 I--"' ~ j SPSI \

3 .....

~ ::.:::-V ==-- I PSI

~ ~

I e

/

1

a

e

0 0 2 4 ~ • 8 10

OFFSET - FEET

RIGHT DUAL

I 4.5 FT SPACING

~ I

1--J 2 ~ ~

3

7

a

10 0

PLATE 8

l~ -- ~ " ""' ,P$1

£: r-=- I ~ _::::; ' '-10 PSI

·~ ' L-...

f'.-s PSI { '-.,

/

/ /

2 3 4 5 $ OFFSET - FEET

~ -IPS/

e 10

l--.1.0'1 RIGHT DUAL

I PSI 0 b.

~.1.- 1fj 6.0 FT SPACING

~""is ~ ~

1-lol lol ... I l:

li: ... 0

I

2

3

4

8

9

10 0

!-~~

2

3

7

8

9

10 0

~ ' ;(,"A ?{: v~ ,, i' I

f.-7 SPSI 1 '-.....1::. p ft =-!PSI

I /I

I

4 5 • 8 10 OFFSET - FEET

RIGHT DUAL

I J 7.5 FT SPACING

t-- r,;, j::,--};;,4 r\ ~ \~ ~ q ,, :p~':(-:1

'-r--~ ~SPSI

/

/

2 3 4 5 6 OFFSET - FEET

~ f't..

~ I P5l

_) /

8 10

NOTE: SOLID LINES ARE TEST DATA; DASHED LINES ARE THEORY. OFFSET MEASURED FROM CENTROID OF LOADED AR£A ALONG X- AXIS.

TAI\EN DIRECTLY FROM R£PORT ON HOMOG£ii£0US CLAYEY- Stt.T TEST S£CTION

ISOBARS OF STRESS MAXIMUM SHEARING STRESS-Tt.tAx

60,000-LB LOAD

OFFSET IN FEET 0.008 7 6 5 4 3 2 I 0 I 2 3 4 5 6 7 8

"' UJ I

0.01

0.02

~ 0.03

~

5 0.04

i= (J

~ 0.05 "UJ 0

0.06

0.07

0.08

0.00 </) w I

~ 0.01

~

5 0.02

i= (J

~ 0.03 "w 0

0.04

r~ r-- ... k r--i"""- ....... ;, ./ / v

~ i_ v ~\ IJ I

'\1 If

I I I

lJ I 11 r i I :\ 'I \

\ \

j J

0.5-FOOT DEPTH

OFFSET IN FEET 87654321012345678

~· r.... -~-.-

....... ~ 1,..... 1/ ......

'\:r" v 1\: /

~\ / _\ L' v ~ ~~ /

2.0-FOOT DEPTH

OFFSET IN FEET <I) 8 7 6 5 4 3 2 I 0 I 2 3 4 5 6 7 8 w 0.00 I (J z ~ 0.01

z 0 i= 0.02 (J w ..J u. w0.03 0

~

082753A

....., :::.:..:

~

l=o., ..... ~ r.;:::: l""'o -

I

6.0-FOOT DEPTH

OFFSET IN FEET 0.008 7 6 5 4 3 2 I 0 I 2 3 4 5 6 7 8

0.01

</) 0.02 w I

~ 0.03

5 0.04

i= u ~005 u. w 0

0.06

0.07

0.08

</) 0.00 w I

~ 0.01

z 5 002

i= u ~ 003 u. w 0

0.04

t:- ..... l,~ ... -......... ~ v / f-"""

'\; /._ r\\ I I ~ 1/

11 I '\\ J 1

" I I ~ li/1\ , \ \) J

1.0-FOOT DEPTH

OFFSET IN FEET 87654321012345678

1:-· t-. ....... .......

""' -v· ..,..--, ·"' ~

" "· "~ .... ·' r--

3.0-FOOT DEPTH

LEGEND

DUAL LOAD DEFLECTIONS

SINGLE LOAD DEFLECTIONS

--

NOTE 250-SQ-1N. PLATE, 3.0-FT DUAL SPACING

SINGLE LOAD DEFLECTIONS WERE INCREASED BY RATIO TO MAKE MAXIMUM DEFLECTIONS FOR SINGLE AND DUAL LOADINGS EQUAL.

POISSON'S RATIO= 0 3

MODULUS OF ELASTICITY= 18,000 PSI.

SURFACE LOAD= 100 PSI.

COMPARISON OF SINGLE AND DUAL

DEFLECTION PROFILES THEORY

PLATE 9

TEST DATA THEORY DEFLECTION IN INCHES DEFLECTION IN INCHES

.004 ooe 001 002 003004 ooe o1o 001 0.4

002 o.o3 oo4 ooe o 10 020030

.... "' "' ... ! :z: .... ... "' 0

0.4

0.~

0.8 0.7 0.8 0.9 1.0

2.0

3.0

4.0

s.o

e.o

001 0.4

0.!>

0.8 0.7 0.8

ti ?:~ "' ... !

~ 2.0

"' 0

3.0

4.0

!>.0

8.0

001 0.4

0.5

0.8 0.7 o.a

:;; ~:~ "' ... ! :z: ti: 2.0

:!! 3.0

4.0

5.0

jlQ

./:

~ ~ ·'

7 / /. /

/ ./

t/

0.!>

0.8 0.7 0.8

tin "' ... ! :z: t: 2.0

"' 0

3.0

4.0

!>.0 6.0

/ / I

v .' /

/ , / /

250-SQ-IN. PLATE -DUAL SPACING 2.5 FT

DEFLECTION IN INCHES 002 003004 006 010 0 20 0 30

/ /

,,.Yv v / ,.

l/ ,, L .L':

001 0.4

0.!>

0.8 0.7 0.8

.... 0.0 UJ 1.0

"' ... ! :z: t:: 2.0

"' 0

3.0

4.0

!>.0

6.0

DEFLECTION IN INCHES 002 003 004 008 010

I

v I

/ /

/

500-SQ-IN. PLATE-DUAL SPACING 3.0 FT

DEFLECTION IN INCHES 002 oo3 0.04 ooe o 10

J 17 I

020 030 001 0.4

0.!>

o.e 0.7 0.8

DEFLECTION IN INCHES 002 003 004 006 010

0 20 030

020 030

I I I I

;' tin "' ... ~

v,

./:

Ll v ;

/ l,.

L

:z: ~ 2.0

"' 0

3.0

4.0

!>.0

8.0

1000-SQ-IN. PLATE-DUAL SPACING 4.5 FT

LEGEND - SINIOLE ~--o DUAL

_I

ll

NOT£' THEORY

08275310

PLATE tO

POISSON'S RATIO= 0.3 MODULUS OF ELASTICITY= 20,000 PSI SURFACE LOADING= 100 PSI

TEST DATA

HOMOCOENEOUS SAND TEST SECTION SURFACE LOADINIO=IOO PSI

SINGLE AND DUAL LOAD DEFLECTIONS

UNIFORM CIRCULAR LOAD

"'0 r ~ f'l1

0 -0.02 ....__

, "'

0.00

0.02

~ 0.04

~ ! 0.01

! ;::: o.oa u "' il. 0.10

"' Q

0.12

0.14

0.18

\

\ \

)

0)-0.02 ° "' :--:z: u ! 0.00 ! ~ 0.02 ;::: ~ 0.04 ..J ... ll: 0.01

~

Ol'fSI!T IN FEI!T 23454 749

C DUAL -1-

I I 1'-~

I I If ~ I I

' • I N I

f sr 'NfiLf

0.5-FOOT DEPTH

OFFSET IN FEET 2 3 4 s • 1 8 9

(DUAL ~

..,....:::: '""7 /:

:/ v

-== -f IN~tE

3.0-FOOT DEPTH

NOTE: IOOQ-SQ·IN. PLATE, 4.5·FT DUAL SPACING.

SINGLE LOAD DEFLECTIONS WERE INCRI!ASED BY RATIO TO MAKI! MAXIMUM DEfLECTIONS FOR SINGLE AND DUAL LOAOINCOS EQUAL ..

DI!I'LECTIONS WERE AVERACOI!D I'ROM THOSE PRODUCED BY 15-,3Q-AND fO-PSI LOADINGS.

0827~B

, "' :z: u ! ~ z 0 ;::: u "' ...1 ... "' a

0 -0.02

r-0.00

0.02

0.04

o.oe

o.oe \ 0.10

0.12 ~/

0.14

O.lt

~-0.02° :z: -u z 0.00

~

~ 0.02 --::-:-

t ~ 0.04 ... "' a o.o4

OFfSET IN FE!T 2 3 4 s 4 1 a 9

(DUAL

1/ ~ 7

I : I

! 7 ~ i v t\ }/

'YN(;L(

1.0-FOOT DEPTH

OFFSETIN FEET 234S 8781

(DUAL ~ -- ;;;;;;;;;'

~ fj'?

...._ f liNfiLIE

4.0-FOOT DEPTH

0 -0.02

r--0.00

0.02 , ... :z: 0.04 u ~

~ o.o8 z 0

It---~

;::: o.oa v

"' ...1 0.10 ... "' 0

0.12

0.14

0.18

011-002° "' . r--:z: v ! 0.00

~ ~ 0.02

P"--t ~ 0.04 ... "' o o.oe

OFFSET IN FEET 2 3 4 s 8 1 a 9

(DUAL

7 ~

Jl r\.. v ·~ ~ 1-f ~IN(; E

2.Q-FOOT DEPTH

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LEGE NO COMPARISON OF SINGLE AND DUAL

DEFLECTION PROFILES DUAL LOAD DEI'LI!CTIONS SINGLE LOAD DEFLECTIONS TEST DATA

SAND TEST SECTION

SINGLE PLATE LOADS

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A 15-PSI LOAD o 30-PSI LOAD D 60-PSI LOAD • ALL LOADS

--- THEORETICAL. POISSON'S RATI0=0.3

THEORETICAL, POISSON'S RATIO=O.S

NOTE: OFFSET MEASURED FROM CENTROID Of LOADED AREA ALONG X-AXIS.

E

PLOTS SHOW MAXIMUM SHEAR STRESSES DERIVED FROM MEASURED NORMAL STRESSES.

PLATE 12

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PLATE 13

CALIP'ORHIA BEARING RATIO-PER CENT 2 • 4 s a 10 IS 20 30 40 so eo 100

0

10 I:="

v: ~-?' f"

/ 20

/

II' 30

1/ v 40

//

eo

70 II .. ! ao ! , to .. .. ~ -.. z .. 2 .. ~ ~

:1 .. z " u i: .. .. .. z ii

B-36 ASSEMBLY LOAD OF 150,000 LB

2 0

10

a 10 IS 20 so 40 so ao 100

f-":

~-"""

~ 20 v~ -

30

40

50

eo

70

ao

eo

100

// /

//

8-36 ASSEMBLY LOAD OF 200,000 LB

LEGENQ

• INADEQUATE

~ BORDERLINE

0 ADEQUATE

NOTE: SOLID CURVES INDICATE PRESENT DESIGN CRITERIA; DASHED CURVES INDICATE PROPOSED DESIGN CRITERIA.

POINTS INDICATE PAVEMENT BEHAVIOR.

0909.S3A

PLATE 14

CALIFORNIA BEARING RATIO-PER CENT 2 3 4 5 8 IO IS 20 30 40 50 80 100

0

10 ~ ~

20 #

30 1/

40 /

50 ;;'

ao

70

ao

eo

100


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10 .-.1;:::'

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30 r

~ lR

40

50 /

ao 1.1

10

ao

eo

100


PLATE 17, MULTIPLE WHEEL REPORT N0.1, MODIFIED TO SHOW PROPOSED DESIGN CURVES

DESIGN THICKNESSES BASED ON

VISUAL OBSERVATIONS

<I) ...

10

20

30

40

r 50 u ?; I

CALIFORNIA BEARING RATIO-PER CENT 4 5 6 7 8 10 15 20 30 40 50

--_.-; ;::::::;...-

0 ..... ~

/......-_~ ,,fi

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/

/ 1/1/

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X 40 COVERAGES ~ 60 <

I / 0 1000 COVERAGES r-

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0

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30

40

50

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B-36 150,000-LB ASSEMBLY LOAD 140-PSI TIRE PRESSURE

UNITS 1,2, AND 3

CALIFORNIA BEARING RATIO -PER CENT 4 5 6 7 8 I 0 15 20 30 40 50

v 1---'

/ v ~....-....-

Vx,........-v v /

/ /, 0

/

v 1/

VI I 1/ /

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I l 6 200 COVERAGES 70

1/ 80

0 1000 COVERAGES 0 2000 COVERAGES


UNITS I, 2, AND 3

NOTE: SOLID CURVES INDICATE PRESENT DESIGN CRITERIA; DASHED CURVES INDICATE PROPOSED DESIGN CRITERIA. POINTS INDICATE THICKNESS REQUIRED

~~~iu~UJ6A~~NE6J~~7~t~~t~~T~g~~' BENEATH SINGLE AND MULTIPLE LOADS.

----

CALIFORNIA BEARING RATIO- PER CENT 80 4 5 6 7 8 10 15 20 30 40 50 80

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80

10

20

30

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50

60

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LEGEND

X 0 COVERAGES 0 1500 COVERAGES 0 2000 COVERAGES


UNITS 4, S.AND 6

CALIFORNIA BEARING RATIO- PER CENT 03 4 5 6 7 8 10 15 20 30 40 50

f-

r-

'-

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X 0 COVERAGES 0 750 COVERAGES

-70

0 2000 COVERAGES f--

80 B-50 100,000-LB ASSEMBLY LOAD

190-PSI TIRE PRESSURE

UNITS 4, 5, AND 6

PLATE 21, MULTIPLE WHEEL REPORT NO.I, MODIFIED TO SHOW PROPOSED DESIGN CURVES

80

DESIGN THICKNESSES BASED ON

EQUIVALENT SINGLE-WHEEL LOADS

0909538

PLATE 15

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CJ)

3 0

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60

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0827531<.

CALIFORNIA BEARIN(; RATIO

4 5 6 7 6 I 10 15 20 30 40 50 60 70 80 10 100

-~--~"" k:::=' ~

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... ~

/.~ A ~ .,

'

50,000-LB DUAL 37.5-IN. SPAC lNG

360-SQ-IN. TIRE PRINT i

I I

CALIFORNIA BEARING RATIO 4 5 6 7 8 9 10 15 20 30 40 50 60 70 80 90 100

-~ ""' ..... ~ ~ F=='"'"

~

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;A 315-IN. SPACING

~sot~t·, T;R~ ~~~~~

I I I

3 0

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10

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CALIFORNIA BEARfN(; RATIO

4 5 G 7 8 !I 10 15 20 30 40 5o eo 10 80 to 100

I~ ~

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r-.......:::

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./

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360-SQ-IN. TIRE PRINT

LEGEND CURVES BASED ON CURRENT CRITERION

----- CURVES BASED ON PROPOSED CRITERION

NOTE: POINTS INDICATE THICKNESS REQUIRED FOR EQUIVALENT SINGLE-WHEEL LOAD, COMPUTED BY EQUATING DEFLECTIONS BENEATH SINGLE AND MULTIPLE LOADS

DESIGN THICKNESS BASED ON

EQUIVALENT SINGLE-WHEEL LOAD MARIETTA

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3 0

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082753J

CALIFORNIA BEARING RATIO 4 s e 1 8 11 10 15 20 30 40 so eo 10 80 110 100

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CALIFORNIA BEARING RATIO 4 5 e 1 e 11 10 15 20 3o 40 50 so 10 80110100

1--..::

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CALII'ORNIA 8EARIN<O RATIO 4 s e 1 8 11 10 15 20 30 40 50 eo 10 80 110 100

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LEGEND CURVES BASED ON CURRENT CRITERION

----- CURVES BASED ON PROPOSED CRITERION

NOTE: POINTS INDICATE THICKNESS REQUIRED FOR EQUIVALENT SIN<;LE WHEEL LOAD, COMPUTED BY EQUATING DEFLECTIONS BENEATH SINGLE AND MULTIPLE LOADS.

DESIGN THICKNESS BASED ON

EQUIVALENT SINGLE WHEEL LOAD STOCKTON TEST 2

"tJ r ~ fT1

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1.0-

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082753L

~

60,000 -LB DUAL 3.0- FT SPACING 1-

500-5Q-IN. PLATE

LEGEND

~

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40

CURVES BASED ON CURRENT CRITERION ----- CURVES BASED ON PROPOSED CRITERION

CALIFORNIA BEARING RATIO 3 4 5 6 7 8 9 10 15 20 30 40 50 60

,{' f/

~ F:== r-~ 1..-.;:

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DESIGN l'HICKNESS BASED ON EQUIVALENT SINGLE-WHEEL LOAD

HOMOGENEOUS CLAYEY-SILT TEST SECTION

3.0- AND 4.5- FT SPACINGS

'1)

r ~ rn

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CALIFORNIA &fARING RATIO

3 4 !> 6 7 8 9 10 15 20 30 40 50 60

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CALIFORNIA BEARING RATIO 3 4 5 6 7 8 g 10 15 20 30 40 50 60

l.,..-..ool ~ ~ -~--

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LEGEND

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:z: .... 0 ... '!: 30

5 "' a:

40

40

,.

NOTE: POINTS INDICATE THICKNESS REQUIRED FOR EQUIVALENT SINCOL£-WHEEL LOAD, COMPUTED BY EQUATINCO DEfLECTIONS BENEATH SINCOL.E AND MULTIPLE LOADS

CURVES BASED ON CURRENT CRITERION ----- CURVES BASED ON PROPOSED CRITI!RION

-U) 082753M


3 4 5 8 7 8 g 10 15 20 30 40 50 60

~ --~

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DESIGN THICKNESS BASED ON EQUIVALENT SINGLE-WHEEL LOAD

HOMOGENEOUS CLAYEY-SILT TEST SECTION

6.0- AND 7.5- FT SPACINGS

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0 3 4 !i II 7 8 9 10 15 20 30 40 50 eo

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082753P

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082753N

3

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HOMOGENEOUS SAND TEST SECTION

250- AND 1000- SQ-IN. PLATES

200

150

100

110

80

f 70

;;: 80 ! 0 50 < s ..J -40 ... ... :r 31: 30

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CALifORNIA BEARING RATIO IN PER CENT 3 "' 5 6 7 8 9 10 15 20 30 -40 !)() 80 70 80 90 100

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LEGEND

CURRENT (d/2 AND 2S) CRITERION

CRITERION BASED ON MAXIMUM SHI!AR RATIO

PROPOSED CRITERION BASED ON MAXIMUM DEFLECTION RATIO

082753F

PLATE 22

~HICKNESS IN INCHES

COMPARISON OF DESIGN CRITERIA

B-29 360-SQ-IN. TIRE PRINT

DUAL SPACING 37.5 IN. C-C

APPENDIX A: EXAMPLE OF THE COMPUTATION OF EQUIVALENT SINGLE WHEEL LOAD

1. This appendix provides a detailed example of the method by

which theoretical maximum deflections are developed for single- and

multiple-wheel assemblies and combined to give a relation between

multiple- and equivalent single-wheel loads.

Assume: A dual assembly, 40-in. c-c spacing, 314-sq-in. contact area,

A, each wheel.

Then: radius, r =-Jf = ~3;4 = 10-in.

4o spacing in radii = 10 = 4 radii between duals.

Al

Plate Al, which is taken from reference (5) (see main report), gives de

flections for a single load in terms of deflection factor, F, such that:

where

prF Deflection, w = E

m

p = load intensity,

E =modulus of elasticity. m

The following tabulation of deflection factors is taken directly from

plate Al:

Table Al

Deflection Factors Offset from Center of Single Load Depth Beneath 2 Radii 4 Radii in. Center or 20 in. or 4o in.

0 1.50 0.39 0.19 r or 10 1.06 0.41 0.20

2r or 20 0.67 0.38 0.20 3r or 30 0.47 0.34 0.20 4r or 4o 0.36 0.29 0.20 5r or 50 0.29 0.25 0.19 6r or 6o 0.25 0.22 0.17

A2

2. By the principle of superposition, the deflection beneath one

wheel of the dual loading is equal to that beneath the center of a

single load plus that at 40 in. (4 radii) offset. Also, deflection be

neath the center of the dual assembly is twice that at 20 in. (2 radii)

offset beneath the single. Thus, by adding the outer columns from the

table on the preceding page and by doubling the center column, we arrive

at the following table of deflection factors:

Depth in.

0 10 20 30 40 50 60

Table A2

Deflection Factors Beneath One Beneath

Wheel Center of Dual of Dual

1.69 1.26 0.87 0.67 0.56 0.48 0.42

0.78 0.82 0.76 0.68 0.58 0.50 0.44

3. The maximum deflection beneath one wheel of the dual represents

the maximum deflection anywhere beneath the dual loading for shallow

depths. Similarly, the maximum deflection midway between the dual wheels

represents the maximum deflection anywhere beneath the dual loading for

deep depths. The maximum deflection beneath the dual wheels in the

transition zone is most easily determined by plotting curves from the

data in table A2 on a single plot and visually adding a limiting, or

transition, curve. It could be determined more exactly by superposing

deflections beneath the individual wheels of the duals for all offsets

between the wheels and selecting the maximum, but the added accuracy

does not justify the i.ncreased effort. Table Al lists deflection factors

beneath the center of a single-wheel load. These are the maximum de

flection factors for a single load. Plate A2 gives maximum deflection

factors for the dual load.

4. The load on a single wheel of the same contact area as one

wheel of the dual assembly that produces a maximum deflection equal to

that beneath the dual assembly is assumed to be equivalent to the

dual loading (refer to part IV of the main report). We may,

therefore, equate deflections from table Al and plate A2. These are

A3

expressed as deflection factors such that w prF =y By using subscripts

m

s and d to denote single and dual, we may write:

w s

And since ws is to equal wd' and rs is to equal rd (this is true since

As is to equal Ad),

Since contact area is the same for both single and dual, the ratio of

total load must be the same as that for unit pressure. Therefore

p Fd s -= F pd

Thus, the ratio of the equivalent single-wheel load to the s

load on one wheel of the dual assembly is the inverse of the ratio of

the maximum deflection factors. In the following table the ratios of

dual- and equivalent single-wheel loads are determined for the various

depths:

Table A3

Load Ratio Depth Single-wheel Dual-wheel Single-to-one Single-to-dual in. Deflection Factor Deflection Factor Wheel of Dual Assembly

0 1.50 1.69 1.13 0.565 10 1.06 1.27 1.20 0.600 20 0.67 0.89 1.33 0.665 30 0.47 0.70 1.49 0.745 40 0.36 0.58 1.61 0.805 50 0.29 0.50 1.72 0.860 60 0.25 0.44 1.76 0.880

A4

The ratios listed in the right-hand columns of table A3 can be applied

directly to the load on the dual assembly (or on one wheel of the as

sembly) to determine the equivalent single-wheel load for the assembly

for the pertinent depth. For example, assume that the dual assembly is

loaded with 50 kip and we are concerned with a depth of 20 in.: From

table A3 the ratio of single- to dual-assembly loads is 0.665; therefore,

the equivalent single-wheel load is 50 x 0.665 = 33.3 kip. Or, we may

use the load on one wheel of the dual which is 25 kip. From table A3

the ratio of single load to the load on one wheel of the dual is 1.33.

The equivalent single-wheel load is, therefore, 25 x 1.33 = 33.3 kip.

The ratios used to relate the 50-kip dual to its equivalent, 33.3-kip,

single-wheel load, are valid for all loadings on this dual assembly.

Thus, the equivalent single-wheel load for the 20-in. depth for any load

can be established.

5. From the 33-3-kip equivalent single-wheel load and the single

wheel CBR curves, the CBR required at a depth of 20 in. to support the

50-kip dual-wheel load can be determined. For the 100-psi-tire-pressure

CBR curves this CBR would be 8.2, and in the same way the CBR values for

other loads can be established. By repeating this procedure for various

depths, the relation between CBR, thickness of pavement and base, and

load can be established and curves drawn for the dual loading selected

as an example. This operation can then be repeated for other dual

loadings and for other configurations as well.

.._ a: 0 ...

1.90

1.80

1.70

1.60

1.!)0

1.40

1.30

1.20

~ 1.10

z 0 ;:: ~ 1.00

0

::; 0.90

.., ::> .J

"' > 0.80

q

FY-•++-+-+- ,_ : oFt'slr~!J,

o.Joi'=FoFFscr'•.?sor

O.IOf-r- OFFS£T=6.00r

n-t-f-bFFSCT ··BOOr I

0.00 I

w=~ Em

l+ + + t+ ~

H- rt+ t-- -i

OFFSET -OOOr

1 Ff OFFSET "'500r

2r

W =VERTICAL DEFLECT ION IN INCHES

3r

-i

:..±± It-·- +

+r-:-~ t-H--+ -t ··r-+-j-H-H-;+ ~ · · " t±::±::±:i • ;- T . d_

4r

DEPTH

r =RADIUS OF LOADED CIRCULAR AREA IN INCHES Em= ELASTIC MODULUS IN PSI

f =DEFLECTION FACTOR z = DEPTH IN INCHES p =SURFACE CONTACT PRESSURE IN PSI

NOTE: FOR POINTS BENEATH THE CENTER OF

THE CIRCULAR AREA (OFFSET= O.Or) f" = 21 ~: r'

OffSETS MEASURED FROM ORIGIN ALONG X-AXIS.

95095S B

ri t t

b:

+-!::

., 7r 6r

DEFLECTION FACTOR F FOR UNIFORM CIRCULAR LOAD OF RADIUS r

AT POINTS BENEATH THE X-AXIS POISSON's RAT10=0.5

PLATE AI

0 Ul 0 00 Ul Ul

>

fl) 1&1 :I: 0 z z :I: ... Q.

"" 0

10

20

30

40

50

eo

DEFLECTION FACTOR 25 75 100 125 150 175 200

' MAXIMUM DEFLECTION

~ BENEATH CENTER -

OF ASSEMBLY ---II

} V/ MAXIMUM DEFLECTION

BENEATH ONE ~WHEEL OF ASSEMBLY

II I~ #

/ ~

~ /

/ -MAXIMUM DEFLECTION

BENEATH ASSEMBLY

NOT£: 40" C-C DUAL SPACING J/4-SQ-IN. CONTA~T AREA

MAXIMUM DEFLECTIO N LOAD BENEATH DUAL WHEEL

analysis of existing data · a. report of the development of b-29 design criteria (4).* b. report...

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