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
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
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
MULTIPLE-WHEEL LANDING GEAR ASSEMBLIES
ANALYSIS OF EXISTING DATA
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
WHEEL LOAD 150,000 LB
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
WHEEL LOAD 100,000 LB
CJ'z -PSI 20 40 60 80 100 120 140
~V /
~ 00
Vc 0
ko I
1 J
WHEEL LOAD 200,000 LB
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
OPFSET IN I' I! ET 2 3 4 s 4 1 a 9
( D'IAL .-II _.,. It"""-
~.r '-{ SINfiLE
5.0-FOOT DEPTH
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
1&1 a: :::> ~ .., a: Q.
1-u < !z 8 1&1 u ~ a: iil ... 0 1-z .., u a: .., Q.
z U)
:3 a: t;
I
.J84 120
110
100
90
80
70
eo !>0
4
30
20
10
0
-10 0
I -1000 SQ-IN PLATE
"\
"-\
., .._
23456789 OFFSET IN FEET
4.5-fT SPACING
oil
0
0
13
12
II
10
9
8
1
6
!>
4
0
0
0
0
0
0
0
I I 1000-SQ-IN. PLAT
......... 3
2
0
o r rJ
', ----0
-I 0 0
~ ~ ,_
23456769 OFFSET IN FEET
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
I 2 z zl I !>Oo-SQ IN PLATE -
1----4 -\1
\ '1~
0 23456769 OFFSET IN FEET
DUAL PLATE LOADS 3.0-FT SPACING
.,,.. I 500-SQ-IN. PLATE
:._..... 1"\. t~- '\ ~ ' ~-
~ 2!>0-SQ-IN. PLATE
..... '\
0
'\
23456769 OFFSET IN FEET
2.5-FT SPACING
I I 250-SQ-IN. PLATE
b. r __ , r\ ~"'-
0 23456769 0 2 3 4 5 6 7 8 9 OFFSET IN FEET
13
12
II
0
0
0
0
0
0
10
9
8
7
6
5
0
4
3
2
0
0
0
0
0
0
-1 0
OFFSET IN FEET
llli l
a.--: :.--.. ,r- - rt. !7 ~
:PI'/
6.0-fT SPACING !>00
r....
SQ P ATE -IN. L
T AXIIi DIRIIC'l'LY Fl!£11 IIIPORT 011 l!ai<XWIIOUS SA!IIl TIST SiCTIOII
STRESS VS X-OFFSET DISTANCE
TMAx AT 0.5-FT DEPTH
0123456789 OFFSET IN FEET
SINGLE PLATE LOADS
IIJ a: :::>
::l IIJ a: Q.
.... u <( .... z 8 IIJ u ~ a: :::> V) ... 0 .... z w u a: w Q.
z V)
:3 a: I;;
I .foS I§
12
II
100
!"\ 90
80
70
60
50
40
30
20
10
0
-10 0
~ 1000 SQ IN PLATE - -
1 t--..
23456789 OffSET IN FEET
4.5-FT SPACING
~-~~ !_1000-SQ-IN. PLAT 130
120
110
100
90
80
70
60
50
40
/I tf
N 1\
30
20
10
0
1
-to 0 " 2 3 4 5 6 7 8 9
OFFSET IN FEET
6 15-PSI LOAD o 30-PSI LOAD o 60-PSI LOAD • ALL LOADS
--- THEORETICAL
NOTE; OFFSET MEASURED FROM CENTROID Of LOADED AREA ALONe;; X-AXIS.
TAKEN DJRrel'LY 1'11011 REPORT 011 llllllGI!IIEOOS SAIID 'l'EBT SECTIOII
E
0.$27~2 A
taw a 500 SQ IN PLATE - -
\ \
0
\ \ 23456789
OffSET IN FEET
DUAL PLATE LOADS 3.0-FT SPACING
! j 500-SQ-IN. PLATE
I
\ i'-
~ 250-SQ-IN. PLATE
\ I
0
1\
23456789 OFFSET IN FEET
2.5-FT SPACING
- 1 T 250-SQ-IN. PLA E
...... \
I \ \
0 23456789 0 2 3 4 5 6 7 6 9 OFFSET IN FEET
0
0
13
12
II
10 0
0 9
8
7
6
5
0
0
0
0
4
3
2
0
0
0
0
-I 0
OFFSET IN FEET
1-:lllill iJI 6.0-FT SPACING
50o-SCHN PLATE z I
I \ I I
I \ I \
~!Oi /y
X.__ I
-------- I
Ill ----------X I I
/ I I z
STRESS VS X-OFFSET DISTANCE
0Z AT 0.5-FT DEPTH
0 23456789 OFFSET IN FEET
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
B-29 ASSEMBLY LOAD OF 70,000 LB
2 3 4 5 e 10 15 20 30 40 so eo 100 0
10 .-.1;:::'
,.,...,.::;:::: ,.-;; ·-fi
30 r
~ lR
40
50 /
ao 1.1
10
ao
eo
100
B-50 ASSEMBLY LOAD OF 100,000 LB
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
V/ / I//
/
/ 1/1/
/I ;/ LEGEND r-
X 40 COVERAGES ~ 60 <
I / 0 1000 COVERAGES r-
/ 0 2000 COVERAGES r-II)
0
~ 70
f-z w :::1 ... ~ 4-0 <I) <I)
~ 03 :<: ~ r f-- tO 0 ... z iii :::120 0 u
30
40
50
60
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/ /
~ X 0 COVERAGES
I l 6 200 COVERAGES 70
1/ 80
0 1000 COVERAGES 0 2000 COVERAGES
B-36 200,000-LB ASSEMBLY LOAD 198-PSI TIRE PRESSURE
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
"""" !::="i"'
80
10
20
30
40
/ 1'
50
60
70
__.-:~ ~
/co
~
~
1,1 /
LEGEND
X 0 COVERAGES 0 1500 COVERAGES 0 2000 COVERAGES
B-29 70,000-LB ASSEMBLY LOAD 100-PSI TIRE PRESSURE
UNITS 4, S.AND 6
CALIFORNIA BEARING RATIO- PER CENT 03 4 5 6 7 8 10 15 20 30 40 50
f-
r-
'-
10 1.---:: ;;::::F ..........-:: v:-
........-::: ;:-./ ~ vo 20
30 _,/
/
~ fj· 40
50 If!
/;
60 i! LEGEND -
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
"0 r > -1 [11
CJ)
3 0
"' "' :r:: \)
10
! 20
:!:
"' "' LIJ z " 30 \)
x .... 0 w a: 5 40 a ... a:
"' ... :r:: \)
50
60
z 10
~
"' "' LIJ z " 20 \)
r .... 0 ... a: - 30 a w a:
~
~
40
0827531<.
CALIFORNIA BEARIN(; RATIO
4 5 6 7 6 I 10 15 20 30 40 50 60 70 80 10 100
-~--~"" k:::=' ~
~II"
~ ~ ~ .......
... ~
/.~ 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=='"'"
~
~ p
~:::: -::;..
~ (-1
40,000- LB OUA L
;A 315-IN. SPACING
~sot~t·, T;R~ ~~~~~
I I I
3 0
"' w l: \)
10
~ 20
!: "' "' uJ z ~ 30
x .... 0 w a: 5 40 0 w a:
50
60
~
CALIFORNIA BEARfN(; RATIO
4 5 G 7 8 !I 10 15 20 30 40 5o eo 10 80 to 100
I~ ~
T ... ~o-~ ~ ~
r-.......:::
~ I-"
./
~,
... ~ ~/'
t ~'}~N.L~P~~~~~
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
"'0 r ~ rn
3 0
<I)
"' :r u
10
! 20
~ ., ., Ill z " 30 u i 1-
Q Ill a: 5 40 0 ... a:
fl) Ill :r u
50
eo
~ 10
~ ., <I) ... z " 20 u i 1-Q Ill a: - 30 a "' a:
v ~I
~ .~
40
082753J
CALIFORNIA BEARING RATIO 4 s e 1 8 11 10 15 20 30 40 so eo 10 80 110 100
~ ~i==F
~ ~ ~ ... I P<t
~ ................. Ill: 0
2 pc Nlrc v /0 "'/
li
I ~~
/ v
/ v
IO~f~~?N_L: :~~NLs~1~~~~M / 3eo-s~N. TIRE PRINT
CALIFORNIA BEARING RATIO 4 5 e 1 e 11 10 15 20 3o 40 50 so 10 80110100
1--..::
-~ ~ - " 2
~ ....... ,· ,
// "' / // 8~000- L 8 DUAL -TANDEM
/ 3 .s-IN. X Be-IN. SPACING
~eo-1so;N.1 T:R~ ~~i~ I I
Ill
ell ... :r u
10
! 20
! ell
"' "' z ~ 30
i: 1-Q
"' ~ 4D 0 ... a:
50
80 7
CALII'ORNIA 8EARIN<O RATIO 4 s e 1 8 11 10 15 20 30 40 50 eo 10 80 110 100
2 PC
I I
7
1.--
~ -;: I-' ,_.. I
~PI I v / ',;"''.
,. ..... .......
~ ~·0 -lt.T V/'
/
l
./ L'
4
v ,, I
II 15fl~-~N.\8 e0~~Ls-Pl~r:!'f,. v 36o-SQ-IN. TIRE PRINT
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
til ... :r u ~ 10
! "' til ... z ~20 :r: r
3 CALIFORNIA I!IEARING RATIO
4 5 6 7 8 9 10 IS ZO 30 40 so eo 1-
.....s ~ v p
II'
~,.
~ ~
30/)00- LB DUAL 3.0- FT SPAC lNG -
"' w J: u 10 ~ ~
"' "' w ~ 20 u J: ....
CALIFORNIA BEARING RATIO
3 4 s· 6 7 8 9 10 IS 20 30 40 so 60
1.0-
1.-.soo" ..--~
~-
~ 0
1-":
~ .;:I-": :;;..-
~ ~" !o~o_o~;~~Ag~~ f-0 ~ 500-SQ-IN. PLATE 0 " 500-SQ-IN. PLATE .., ~30 0 ... 0::
40
L "' ~ 30 0 w a:
40
CALIFORNIA BEARING RATIO
"' ...
3 4 s 6 1 e 9 10 15 zo
~
30 40 50 60
lr--1-f-~
J: v 10 ./:'-. ~ "' "' J:
~ 10 ~ ~ til
"' "' ~20 u :r: ~
0 ... !;3o ~ 0 w 0::
40 /"
~
l .if
~
NOTE' POINTS INDICATE THICKNESS REQUIRED FOR EQUIVALENT SINGLE-WHEEL LOAD, COMPUTED BY EQUATING DEfLECTIONS BENEATH SINGLE AND MULTIPLE LOADS
082753L
~
60,000 -LB DUAL 3.0- FT SPACING 1-
500-5Q-IN. PLATE
LEGEND
~
"' "' "' z :.: 20 ~ I .... 0 .... ~ 30 0
"' a:
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..-.;:
~ ~-A
-::?"" v~
/~r" 60,000 -LB DUAL
/~ 4.5-FT SPACING r-
500-SQ-IN. PLATE
DESIGN l'HICKNESS BASED ON EQUIVALENT SINGLE-WHEEL LOAD
HOMOGENEOUS CLAYEY-SILT TEST SECTION
3.0- AND 4.5- FT SPACINGS
'1)
r ~ rn
., ... :z: u ! 10
~ ., ., "' z :3 20
i 1-0 ... ~30 a "' a:
., ... :z: ~~o
z ., ., ... ~20 u % 1-0 ... !!:30 ::;)
a "' a:
40
CALIFORNIA &fARING RATIO
3 4 !> 6 7 8 9 10 15 20 30 40 50 60
T ..... --~ ~ r:-
~ ... :;.. _,.~
~.,
~ / 30,000- LB DUAL 6.0-FT SPACING -
500....SQ-IN. PLATE
CALIFORNIA BEARING RATIO 3 4 5 6 7 8 g 10 15 20 30 40 50 60
l.,..-..ool ~ ~ -~--
~ ~-.n!.
~ I ~
I
I/~ 60,000-LB DUAL
/v 6.0- FT SPACING -500-SQ...JN. PLATE
/ ~·
;j,.
LEGEND
., ... :z: ~ 10
!: ., <I) ... ~ 20 1.)
: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
CALIFORNIA BEARING RATIO
3 4 5 8 7 8 g 10 15 20 30 40 50 60
~ --~
II"'""
I'll ~
~ ~~
~;.. ~.-· 3 0,000- L B DUAL
7.5-FT SPACING 500-SQ-IN. PLATE
CALIFORNIA BEARING RATIO 3 4 5 6 7 8 9 10 15 20 30 40 50 80
~ .... ~,_;. ~
~ ==-"""
.At-~ ...
~~
....... ~~ 60,000-LB DUAL 7.3- FT SPACING ,_...
v ~?' 500-SQ-IN. PLAT£
v /
DESIGN THICKNESS BASED ON EQUIVALENT SINGLE-WHEEL LOAD
HOMOGENEOUS CLAYEY-SILT TEST SECTION
6.0- AND 7.5- FT SPACINGS
""(J
r ~ J'TI
1\) 0
CALIFORNIA I!IEARING RATIO 3 4 !i e 7 8 9 10 IS 20 30 40 so eo --v ~
~ ~~
~ ~ 110f.OO- LB DUAL
3. -FT SPACING f.-
~ 500-SQ-IN. PLATE
40
CALif"ORNIA BEARING RATIO
0 3 4 !i II 7 8 9 10 15 20 30 40 50 eo
..-1-----~~
~
,?'
~ eo,OOO-LB DUAL
1/-~ 3.0- FT SPACING ~
500-SQ-IN. PLATE
f 40 /
LEGEND
., ... :z: 0 10 ~ ~
:ll ... ~ 20 !:! :z: ... Q ... !!: 30 5 ... a:
ell ... :z:
40
~ 10
! ., ., ... z "20 0 x ... 0 ... 3 30 a "' a:
40
NOTE: POINT$ INDICATE THICKNESS REQUIRED FOR EQUIVALENT SINGLE-WHEEL LOAD, COMPUTED BY EQUATING DEFLECTIONS BENEATH SINGLE AND MULTIPLE LOADS
CURVES BASED ON CURRENT CRITERION ----- CURVES BASED ON PROPOSED CRITERION
082753P
3
~
3
~1
CALIFORNIA BEARING RATIO 4 5 8 7 II 9 10 IS 20 30 40 so eo
I .....
~~ p
'I' ~ ~~ ;
' ' / !
H-: 30,000 -LB DUAL e.O- fT SPACING
500-SQ·IN. PLATE
CALIFORNIA BEARING RATIO 4 5 8 7 II 9 10 15 20 30 40 so eo
/. ~
..-~ ~ ~ ...,..,.
~ ~
~
~ ~
'V~~ eo,OOO-LB DUAL 8.0- FT SPACING 1-
1
' i
500-SQ-IN. PLATE
DESIGN THICKNESS BASED ON EQUIVALENT SINGLE-WHEEL LOAD
HOMOGENEOUS SAND TEST SECTION
500- SQ -IN. PLATE
.,
.... X u ~ 10
~ ., ., .... z ~20 i: t-o .... ~30 a .... a:
40
40
3
~
3
~
CALIFORNIA &EARING RATIO 4 !> 6 7 8 9 10 I!> 20 30 40 50 80
~
..-'!r.=: ~f'=;P
~
15,000-LB DUAL 2.5-FT SPACING
250-SQ-IN. PLATE
CALIFORNIA BEARING RATIO 4 s 6 1 e 11 10 1s 20 30 40 !>0 60 --..--.---
~ ~
~~ ~ :' ~ 30,000-LB DUAL
2.5- FT SPACING ~-SQ-IN. PLATE
LEGEND
., loJ :z: u 10 ~ ~
::! loJ
~ 20 ~ X .... 0 loJ
~ 30 a .... a:
.,
.... X
40
! 10
~ ., ., .... z "20 u l: .... 0 .... 3 30
s a:
40
NOTf: POINT.S INDICATE: THICKNfSS REQUIRfD ,OR fQUIVALENT .SI~E-WHEEL LOAD, COMPUTfD BY EQUATING DEFLECTIONS BENEATH SINGLE AND MUL TIPLf LOADS
CURVES BASED ON CURRENT CRITERION ----- CURVES BASED ON PROPOSED CRITERION
082753N
3
v
3
CALIFORNIA BEARING RATIO 4 !> 6 7 8 II 10 15 20 30 40 so eo
~ ~ 1- lo-' -~ f'?'"
~
~ ~
d~ 60,000-LB DUAL 4.5-f'T SPACING f-
)f 1000-SQ-IN. PLATE
I
CALIFORNIA BEARING RATIO 4 5 8 7 8 II 10 15 20 30 40 50 60
_r.::; I-"'
~ ~ ~
A
.......:: ~ /'
120,000-LB DUAL 4.5- FT .SPACING r-IO~Q-IN. PLATf
,/ I'
DESIGN THICKNESS BASED ON EQUIVALENT SINGLE-WHEEL LOAD
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
20
15
CALifORNIA BEARING RATIO IN PER CENT 3 "' 5 6 7 8 9 10 15 20 30 -40 !)() 80 70 80 90 100
.-11!'1 ~ ~ ~ t::: ,;--10
401000 LB ~ ~ .... ...-! ~ d) ... :r ~20
~ ., ., ... ~30 u
~ 0 ... ~40 a ... a:
50
$0
1/..
~ ~,
l/ /
v if' '('
I/ )
v'
~ ~-1/~~ ~ ASSEMBLY LOAD 70~000 LB
~~· '/ ll~
~~ i~
!)2 I_JljJ1 II ~ cEN1
5 4 .J
"'pE 6 II rro' sV ~,. 10 g E"~'"'G 15 ill, '/ 1/ I 11,.,,,. 8 20 c.4t.IF0 30
40 100 90 80 7'0 60 50
.., / / / , ~ v ./ / 7 , , -z:;. W' rT
./ ./ / / / / / ;--r_ £c'l · rASSEMBLY. _..,... / l/ I/ / / ./--ILM 7-J 'Vi.~ LOAD
L L L. lL / -~ --r t:,.....--"" 'J,... '/ l7'0,000L8
/ / v / v v ...Y7_ --1; 'I 17 I I I j / i/ I [7 I //, 'IJ [l k ~ .... IOii"
I 1/ I J 1/ j /_ II/A rt:; Ft ~ ~40000 LfJ
tf--7 --!.'--li 1--lf . -11 It 1- D ll-1 ~ ~~ rl I I 7 J iJl r rJ
I/ J'!J V!J r; v J
VJ v v I/ v Ll I/ ~ VI v v j
200
150
I 00 SIO
80 ., 70 ~
" &Oz
50~ 0 ..J
40~ Ill ::! ...
30 ::l <
20
I 5
I 0 10 I 2 3 4 5 6 7 8 II 10 15 20 30 40 50 60 70 80
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+ ~
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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