Analysis and Evaluation of the Development of Various k-Values for Use in Design Robert Rodden, Director of Engineering Strategy and Data Science, MEGASLABTM, Marietta, Georgia, USA

Eric Ferrebee, P.E., Director of Technical Services, American Concrete Pavement Association (ACPA), Rosemont, Illinois, USA

For the corresponding author: robert@megaslab.com

KEYWORDS: soil, support, k-value, design, CBR, subbase

Conflict of Interest: None

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1. ABSTRACT

Inconsistency exists between common conversions for soil index properties like CBR to a design k-value and a widespread nomograph that became the definitive reference on the topic in the industry in the United States. Propagation of these inconsistencies into guidance from groups like the American Concrete Pavement Association (ACPA) and American Concrete Institute (ACI) Committees 330 and 360 has contributed to confusion in the industry. Advancements between the pavement and slab-on-ground communities have occurred in parallel but inconsistent of each other, adding more confusion. ACPA developed a conversion set to better align the industry on a static k-value for design; while the ACPA model is included in StreetPave, PavementDesigner.org, and the ACPA App Library, outdated conversion equations are frequently used due to familiarity and lack of understanding of the underlying principles. This paper presents a summary of the industry's prior practices and recommendations, a detailing of the approach proposed by ACPA, and guidance on which k-value is recommended for design of concrete pavements and slabs-on-ground.

2. INTRODUCTION

Much confusion exists in the concrete flatwork (e.g., pavement and slab-on-ground) industry in the United States (U.S.) about which modulus of subgrade reaction (k-value) to use in design. This stems from an over-reliance on distilled guidance relayed from prior efforts and research, standardized design tables, local practice, and a lack of understanding of the underlying assumptions and principles. As so often voiced within the civil engineering discipline, the “we’ve always done it this way” mantra stands up to criticism if the processes produce acceptable results for owners and users of infrastructure. With a focus on optimization and an increasing sophistication of concrete flatwork engineering, engineers conducting such designs need to understand the nuances of the origins and application of k-value.

3. RESEARCH SIGNIFICANCE

Owners interested in lowest initial and life-cycle cost are increasingly adopting innovative materials, construction methods, etc. Often absent from this discussion is the optimization of the design of concrete flatwork. The consideration of the appropriate k-value for use in design is an important factor for an optimized, reliable design that will lead to predictable, risk-managed performance in concrete flatwork.

4. K-VALUE IS NOT A UNIQUE, DEFINITIVE SOIL PROPERTY

The most referenced plate load test in the U.S. is AASHTO T222 / ASTM D1196, “Standard Test Method for Nonrepetitive Static Plate Load Tests of Soils and Flexible Pavement Components for Use in Evaluation and Design of Airport and Highway Pavements”. Using this test, the static k-value is typically determined with a 30 in. (760 mm) diameter plate loaded to achieve 0.05 in. (1.3 mm) of deflection. For example, a 3,535 lb (1,600 kg) load will yield a static k-value of 100 psi/in. (27 MPa/m) per this test, at a contact pressure of just 5 psi (35 kPa). All else equal, if a different plate size is used, the measured static k-value is different, meaning the same soil will have multiple static k-values. Fundamentally, the test presents a deflection resulting from an applied pressure (e.g., inches per psi or meters per MPa) and not a volume per force measure (e.g., pci) as often misleadingly represented in test reports and literature.

Test sensitivities extend beyond the load plate size, with another key variable being the speed of testing. This is easy to understand when considering a non-Newtonian support like quicksand. If an individual stands on quicksand the response is different than if they run on it; stand still and sink but run and stay atop the surface. While this is an extreme example, the nature of the response of a support system to short-period, dynamic loading is different than to long-period, static loading. Falling weight deflectometer tests like ASTM D4694 are used to characterize this dynamic k-value behavior [FHWA 2017], and the difference between static and dynamic has been documented for over 50 years.

AASHTO 1993, the most historically common concrete pavement design procedure in the U.S., contains this simple, rough approximate relationship between static and dynamic k-values:

Static k−value= Dynamic k−value2 (1)

It is important to understand the specifics of the plate load testing setup, the nature of the test being static or dynamic, and details on the data interpretation and manipulation when considering a design k-value. Further, there is rarely a distinction between static k-value and dynamic k-value in communication between the various engineers involved in flatwork projects.

5. COMMON CONVERSIONS ARE IMPERFECT AND INCONSISTENTLY PROPAGATED IN INDUSTRY GUIDANCE

While conversions for soil index properties such as the Resistance Value (R-value) also exist, this paper only discusses the more common California Bearing Ratio (CBR) basis for estimation of soil mechanical properties. As noted in AASHTO 1993:

“The correlation is given by the following relationship:

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M R ( psi )=1,500 ×CBR (2)

The data from which this correlation was developed ranged from 750 to 3,000 times CBR. This relationship has been used extensively by design agencies and researchers and is considered reasonable for fine-grained soil with a soaked CBR of 10 or less…

If the slab is placed directly on the subgrade (i.e., no subbase), the composite modulus of subgrade reaction is defined using the following theoretical relationship between k-values from a plate bearing test and elastic modulus of the roadbed soil:”

k−value ( psi /¿ . )=M R

19.4(3)

where MR = Resilient Modulus of the Subgrade, a measure of the stiffness of unbound material under specific moisture, density, and stress level in a triaxial test, and the k-value is static. Note the interchangeable use of elastic modulus, which is more common for bound material, and resilient modulus which is more common for unbound material in this AASHTO 1993 quote.

While k-value is a design parameter specific to concrete flatwork design, MR is also used in asphalt pavement design. There have been extensive, ongoing efforts to refine the correlation between CBR and MR [e.g., George 2004], with no clear consensus on a most appropriate correlation. The above excerpt from AASHTO 1993 notes the high variability of the data, but a single, constant correlation coefficient was ultimately used in the deterministic design framework of AASHTO 1993.

Appendix HH of AASHTO 1986 provides details on the origin of Eq. 3, specifically that it is theoretical in nature, assumed a 30 in. (760 mm) flexible load plate, despite the plate load test using a rigid plate, and assumed the support to be linear elastic, which is not representative of all roadbed materials.

Application of Eqs. 2 and 3 for a CBR of 3 estimates the k-value at 232 psi/in. (63 MPa/m). This k-value is over twice that which is expected per Fig. 1, a nomograph that has propagated very widely in concrete pavement texts.

By its very nature, MR, as determined by AASHTO T307, is a dynamic property and has been considered appropriate for road pavement design because traffic imposes dynamic loading onto the pavement surface. The resultant k-value from this equation set is a dynamic k-value and to yield a static k-value requires applying Eq. 1, which brings the resultant static k-value for comparison to Fig. 1 down to 116 psi/in (36 MPa/m). Neither the nomograph in Fig. 1 nor the equation set from AASHTO 1993 make this explicitly clear.

Civil engineers tend to methodically apply a design framework upon its standardization. Committee-approved guidance should be trusted and engineers need not delve into the origins of recommendations assuming a practical understanding of the principles. On the topic of AASHTO 1993 conversions, it is worth noting the importance of NCHRP 1972, a guide developed to help in updating the then-evolving AASHO pavement design framework.

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Figure 1. Approximate interrelationships of soil classifications and bearing values (PCA 1984, PCA 1988, ACI 2002, ACI 2010, ACI 2017).

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NCHRP 1972 recommended Figure 2 as a correlation chart; while this nomograph was included in Appendix FF of AASHTO 1986, it was not included in the main body of AASHTO 1993, the design guide used by most practicing engineers. If it had been included, the conversion of Eq. 2 might have met scrutiny earlier. This nomograph shows a CBR of 2 approximately equaling an MR of 3,000 psi, in-line with Eq. 2, but a CBR of 10 (the maximum of the valid range as reported in the qualifying text of AASHTO 1993) is shown to equal less than 9,500 psi, which is far less than the 15,000 psi estimated from Eq. 2.

FHWA 1997 focused on improved characterization of both static and dynamic k-value from new, widely collected field data and proposed new relationships for variables like CBR to static k-value but its guidance did not propagate into common practice.

Because of their simplicity, the AASHTO 1993 set of Eqs. 2 and 3 propagated into most post-2000 industry guidance alongside Fig. 1, which was already a mainstay in concrete flatwork design guidance. This is evidenced by ACI committees 325, 330, and 360 all including some of these concepts in their recent guides. Table 1 details which of these guides includes Fig. 1; Eqs. 2 and 3; or the AASHTO 1993 qualifiers on Eq. 2 as quoted earlier in this paper. It is worth noting that none of these ACI guides includes discussion on the difference between static and dynamic k-value, thus practicing engineers should not be expected to universally know this nuance.

Table 1. Inclusion of Various Support System Conversions Guidance in ACI Design Guides

Document Application(s) Fig. 1 Eqs. 2 & 3 Qualifiers Static vs. DynamicACI 325.12R-02 Streets & Roads XACI 330.2R-17 Industrial Lots X XACI 360R-10 Slabs-on-Ground X X X

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Figure 2. Correlation chart for estimating soil support (AASHTO 1986).

6. MODELING K-VALUE CAN BE FUNDAMENTALLY DIFFERENT

Beyond a physical test and correlation with index soil properties, a k-value is used in the structural design of concrete flatwork. The two most common means of characterization of the support layer under a concrete slab in a structural model are elastic solid or Winkler (Fig. 3). The Winkler model ignores the deflection basin in the support system, instead considering the support as a series of equally engaging springs; while this model was used by Westergaard and is used in some FEM, it tends to underestimate the value of the support system while overestimating deflections of the slab system. The elastic model can capture a non-linear response, soil cohesion, slab/base friction and other complications but is often limited to finite element methods (FEM) because of its complexity.

Winkler Model Elastic Solid Model

Figure 3. Illustration of common foundation deflection models under a uniform load (Skar et al. 2019).

Further, some modern design frameworks such as AASHTOWare Pavement ME (AASHTO 2020) ultimately use a hybrid of these two models through a complex series of equations to develop an effective dynamic k-value through theoretical backcalculation of the system. What this means in practice is that the reported effective dynamic k-value should not be expected to be reproducible directly in the field by either static or dynamic k-value testing because the effective dynamic k-value is a calculated intermediate parameter in a series of equations taking design input values through calculations that include calibration to field performance.

7. DESIGN THICKNESS IS RELATIVELY INSENSITIVE TO K-VALUE

The design concrete slab thickness is relatively insensitive to k-value for ordinary ranges considered. As an example, Fig. 4 shows that increasing the static k-value from a very low (e.g., k-value of 100 psi/in. [27 MPa/m] level such as a clay subgrade) to a very high level (e.g., k-value of 500 psi/in. [136 MPa/m]) might yield a reduction in design slab thickness of up to 1.5 in. (38 mm) by some pavement models for highway traffic. To accomplish such a slab thickness reduction might require a significant investment in the support layers. The contention of industry guidance has been that thickening and stiffening support layers more than necessary for functional reasons is not justified by the increased cost (ACPA 2007).

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The real sensitivity to k-value occurs when the support system is unreasonably soft, such as below a static k-value of 100 psi/in. (27 MPa/m). For this reason, many experts in the industry, including the authors, have suggested ensuring a minimum static k-value of 100 psi/in. (27 MPa/m) as a useful lower boundary in practice. Any increase beyond this should consider not only the cost increase but also the possible concerns for in-service consolidation of an unstabilized base layer (e.g., if there is to be 10% consolidation in-service, it is better to have this occur on just a 4 in. [100 mm] thick layer versus a 12 in. [300 mm] thick layer – see ACPA 2007) and that it has been predicted, against common convention, that if the k-value in the field becomes too great then the risk of top-down cracking due to the interaction of curling/warping and multiple loads also becomes a concern, such as the Phoenix case in Fig. 4 where the design slab thickness must increase above a k-value of 300 psi/in. (80 MPa/m) to maintain performance (Rodden et. al 2014).

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Figure 4. Sensitivity to static k-value by AASHTO 1993, AASHTOWare Pavement ME in Chicago (ORD) and Phoenix (PHX), and ACPA StreetPave (Rodden et al. 2014).

8. STANDARD PRACTICE IN THE U.S.

In the U.S., it is exceedingly uncommon to see plate load testing in the field as part of construction activities. Rather, design values are estimated from soil index properties characterized by geotechnical engineers. Engineers commonly arbitrarily assume a CBR of 3 or a k-value of about 100 psi/in. (27 MPa/m) in designs, regardless of if the design is for a pavement or slab-on-ground, and with no discussion on static, dynamic, or effective k-value. The assumption in these recommendations is that a higher value is expected in the field so this is viewed as a reasonable, conservative design input.

Many geotechnical reports and specifications in the U.S. do provide guidance on proofrolling of the subgrade. The weight and nature of the vehicle vary as does the limit on allowable subgrade displacement. If the limits set in a specification are for 0.25 in. (6 mm) or less deflection (rutting) under a pneumatic tire in a loaded truck passing slowly with a tire pressure of 110 psi (760 kPa), then the apparent static k-value will be 110 psi / 0.25 in. = 440 psi/in. (119 MPa/m). This is an apparent value because the contact area is less than that of a standard 30 in. (760 mm) diameter static k-value testing plate and should be considered approximate as the tire is not a rigid plate. Assuming the tire has a diameter of 10 in. (250 mm), adjustment per a commonly accepted equation (Anyang et al. 2018) would yield an approximate static k-value of 440 psi/in. (119 MPa/m) x [10 in. (250 mm) / 30 in. (760 mm)] = 147 psi/in. (31 MPa/m).

If a concrete pavement design is an alternate to an asphalt design then proper consideration of the support stiffness for the asphalt design and arbitrary assignment of a conservative value in the concrete alternate undermines the competitive environment (e.g., concrete will be made thicker and less competitive than asphalt at the bidding table because the asphalt design might use a realistic MR but the concrete design uses an overly conservative k-value, which often completely ignores the value of subbase layers placed atop the subgrade).

9. ACPA CONVERSION FOR CBR TO MR AND STATIC K-VALUE

Initial versions of ACPA’s StreetPave software included the AASHTO 1993 equations because the design tool was for dynamic, roadway traffic. The contradiction that Eqs. 2 and 3 creates against an expected k-value of 100 psi/in. for a CBR of 3 per Fig. 1 was raised during beta testing of the software. To add credibility, appease critics, include additional conservatism in the design, and meet the expectations of Fig. 1, a set of correlation equations was developed and adopted by ACPA for use in StreetPave and which have been undocumented until this paper.

Fig. 2 served as the basis for the first step of this conversion, with the following conversion from CBR to MR developed by ACPA from the source NCHRP data:

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M R ( psi )=1,941.5 ×CBR0.6845 (4)

Coincidentally, the ACPA conversion is conservative to but of a similar trend to that presented in the Mechanistic-Empirical Pavement Design Guide (NCHRP 2001) that is the basis of the AASHTOWare Pavement ME Design tool:

M R ( psi )=2,555 ×CBR0.64 (5)

Fig. 5 shows these two conversions against that of Eq. 2, with the AASHTO 1993 line changing to a light blue after its reported maximum boundary of CBR of 10.

Figure 5. Comparison of AASHTO 1993, NCHRP 2001, and ACPA conversions from CBR to MR.

With k-value being the support system input for concrete flatwork design rather than MR it was necessary for ACPA to develop a conversion from this estimated MR value to an appropriate k-value. It was decided by the committee developing the StreetPave design software that a static k-value would be used to remain conservative and the output of MR to static k-value correlation would thus match that of Fig. 1. The following equation set was developed by ACPA to produce this desired result:

If MR ≤ 15,090 psi (104 MPa)

k−value (psi /¿ .)=M R( psi)

(1.15 ∙ 10−7 × M R2−4.68 ∙ 10−4 × M R+41.1)

(6)

If MR > 15,090 psi (104 MPa)

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k−value (psi /¿ .)=M R( psi)

(1.06 ∙10−8 × M R2−7.61∙10−4 × M R+69.5)

(7)

Using Eq. 4 with Eq. 6 or 7, notable points of interest include CBR of 3 = static k-value of 100 psi/in.; CBR of 10 = static k-value of 200 psi/in.; CBR of 20 = static k-value of 250 psi/in.; CBR of 50 = static k-value of 500 psi/in.; and CBR of 100 = static k-value of 800 psi/in.; illustrating the ability of this equation set to achieve the most industry-recognized correlation of CBR and static k-value per Fig. 1. This equation set developed by ACPA is included in StreetPave, the ACPA Static k-value Calculator (apps.acpa.org/applibrary/KValue/), other ACPA concrete design software such as WinPAS, and www.pavementdesigner.org.

10. COMPOSITE STATIC K-VALUE

Per the CBR testing method, a high-quality, well-graded and dense compacted stone material should have a CBR of 100, corresponding to a static k-value of about 800 psi/in. (217 MPa/m) per Fig. 1. With the 30 in. (762 mm) diameter plate load test under 0.05 in. (1.25 mm) deflection, this would equate to a pressure of just 40 psi (275 kPa) and a plate load of 28,274 lb (12,825 kg). Somehow, in design, such k-values are not common, even when thick layers of well-graded compacted stone materials are used, because of the ultra-conservative nature of the original, historic guidance that has propagated through time. For example, in ACI 330.2R-17, composite k-value tables show only one value in this 800+ psi/in (217+ MPa/m) range, and that is for a 12 in. (300 mm) thick cement treated or lean concrete base atop a soil with a starting k-value of 200 psi/in. (54 MPa/m), or CBR of 10. It should be expected, though, that this structure is much stiffer than a granular base system.

The ACI 330.2R-17 composite k-value tables reference ACPA 2006, which in-turn was a propagation of PCA 1966, which had its

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Figure 6. Chart for estimating composite k-value (AASHO 1972).

roots in modeling by Burmister 1943. Interestingly, guidance in AASHTO 1972, as shown in Fig. 6, shows composite k-values that are much higher than this series of guidance propagated by PCA, ACPA, and ACI. Fig. 6 shows that even poor subgrade, when overlaid with any subbase at all, will have a composite static k-value over 100 psi/in. (27 MPa/m), in contradiction to ordinary U.S. geotech reports.

The ACPA equation set, as included in StreetPave and www.pavementdesigner.org for the composite static k-value is from FHWA 2006, in-turn developed from guidance documented in an appendix of AASHTO 1986:

ln (k−value( psi¿ .))=−2.807+0.1253 [ ln ( DSB ) ]2+1.062 [ln ( M R ) ]+0.1282 [ln ( DSB ) ] [ ln ( ESB ) ]−0. 4114 [ln ( DSB ) ]−0. 0581 [ ln ( ESB ) ]−0 .1317 [ ln ( D SB ) ] [ln ( M R ) ]

(8)

where DSB is the subbase thickness (in.) and ESB is the subbase elastic modulus (psi).

11. CONSIDERATION IN SLABS-ON-GROUND

Because State Highway Agencies organize with AASHTO, FHWA, universities, and other groups to advance research and models, guidance on the issue of k-value is predominately from the pavement community. With ACI 360R-10 containing Fig. 1 and many slab-on-ground design approaches originating in pavement theory, the norm in the slab-on-ground niche has been to borrow from the pavement community as suitable. On the topic of k-value, applications of pavements have truly static loading under conditions like dolly pads of disconnected trailers resting on the pavement surface, but the static k-value is used in most pavement design approaches to maintain conservatism while only a few advanced design models like AASHTOWare Pavement ME use an effective dynamic k-value. In slabs-on-ground, dynamic loading that could consider a dynamic k-value includes traffic like that from lift trucks and static load that could consider a static k-value includes rack legs and similar non-moving loads.

While some have suggested that subgrade and support settling be included in the composite k-value (Walker and Holland 2016) others have pushed back against this practice (Anyang et al. 2018), noting that the k-value “is a stiffness parameter and must be estimated at deformations in which the load-deformation curve is generally linear.” The assertion behind this mindset is that settlement-related concerns need to be resolved by a geotechnical engineer and the slab should not be thickened to counteract such concerns unless the slab is instead considered as a structural slab per ACI 318.

Further, Walker and Holland 2016 provides no simple means by which a practicing engineer can apply the logic behind a long-term static k-value that considers settlement if the supporting soils are clay. For non-clay soils and the soft clay (with modulus of 2,000 psi [14 MPa]) as presented in Walker and Holland 2016, Eq. 6 or 7 by ACPA are conservative to the recommended design value. If an engineer must reduce soil static k-value for a slab-on-ground application out of

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concerns for settlement because the geotechnical engineer cannot resolve the issue, the Eq. 6 or 7 with a safety factor of 0.65 applied provides design static k-value results conservative to those documented in Walker and Holland 2016 for clay soils with a modulus over 2,000 psi (14 MPa) (e.g., a modulus of 4,800 psi [33 MPa] gives a static k-value of 116 psi/in. [31 MPa/m], which adjusted by 0.65 equals 75 psi/in. [20 MPa/m]). This static k-value should then be considered that of the soil and any subsequent additional layers would add to a composite static k-value as calculated per ACPA.

12. DISCUSSION AND RECOMMENDATIONS

The pervasive nature of Fig. 1 in concrete pavement and slab texts set a strong expectation on static k-value for a given CBR or the other soil index properties or classifications shown. Despite this figure showing static k-value of the highest quality soils exceeding 500 psi/in. (136 MPa/m), the conservative nature of civil engineers when developing design tables looked to downgrade this value much more than necessary. Further, addition of a subbase was shown to greatly increase composite static k-value greatly per AASHO 1972 and it has long been known that dynamic k-value is approximately twice that of static k-value. Practice in the U.S. has not considered the nuances of all the guidance that exists on k-value, meaning that it does not usually account for the value of support layers under concrete flatwork.

If a designer is using a modern concrete pavement design framework in which dynamic loads might consider an effective dynamic k-value where that value is an intermediary in a series of steps to consider well-calibrated performance predictions, it is recommended that this advanced approach be used so long as the designer understands that there is a different meaning between this k-value and what might be measured in either static or dynamic k-value tests in the field.

For all other design cases, with old and modern design frameworks alike, it is recommended that the ACPA static k-value and composite static k-value be used to maintain a balanced conservatism with a realistic k-value. For a design framework like StreetPave or pavementdesigner.org, this will prove conservative because the k-value in such applications of roadways should always be dynamic, as the loads are, rather than taking a static assumption. For slabs-on-ground where loads like lift truck traffic is dynamic but the design framework might be FEM, the same holds true. For static loads, such as dolly pads on pavements or rack legs, again a static k-value condition with design frameworks as simple as Westergaard or yield line is warranted, and the value is also appropriate when using more realistic FEMs to mode the complexities of square base plates on rack legs and interactions of such legs and lift truck traffic nearby.

13. REFERENCES

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AASHO 1972, Interim Guide for Design of Pavement Structures, American Association of State Highway Officials.

AASHTO 1986, Guide for Design of Pavement Structures, American Association of State Highway and Transportation Officials.

AASHTO 1993, Guide for Design of Pavement Structures, American Association of State Highway and Transportation Officials.

AASHTO 2020, AASHTOWare Pavement ME Design, https://me-design.com/MEDesign/.

ACPA 2006, Design of Concrete Pavement for Streets and Roads, IS184, American Concrete Pavement Assocaition.

ACPA 2007, Subgrades and Subbases for Concrete Pavements, EB204, American Concrete Pavement Assocaition.

Anyang M.Y., Atarigiya, B.D., Ofori-Addo, R., and Allotey, N.K. 2018, Plate Load Test: Getting it Right, 49th GHIE Annual Conference.

Burmister, D.M. 1943, The Theory of Stress and Displacements in Layered Systems and Applications to the Design of Airport Runways, Proceedings of the Highway Research Board.

FHWA 1997, LTPP Data Analysis, Phase 1: Validation of Guidelines for k-Value Selection and Concrete Pavement Performance Prediction, FHWA-RD-96-198, Federal Highway Administration.

FHWA 2006, Geotechnical Aspects of Pavements Reference Manual, NHI-05-037, Federal Highway Administration.

FHWA 2017, Using Falling Weight Deflectometer Data with Mechanistic-Empirical Design and Analysis, FHWA-HRT-16-011, Federal Highway Administration.

George, K.P. 2004, Prediction of Resilient Modulus from Soil Index Properties, The University of Mississippi.

NCHRP 1972, Evaluation of AASHO Interim Guides for Design of Pavement Structures, Report 128, National Cooperative Highway Research Program.

NCHRP 2001, Guide for Mechanistic-Empirical Design of New and Rehabilitated Pavement Structures, Appendix CC-1: Correlation of CBR values with Soil Index Properties, National Cooperative Highway Research Program.

PCA 1984, Thickness Design for Concrete Highways and Street Pavements, EB109, Portland Cement Association.

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PCA 1988, Design of Heavy Industrial Concrete Pavements, IS234, Portland Cement Association.

Rodden, R., Voigt, G., and Wathne, L. 2014, Comparison of Roadway Jointed Plain Concrete Pavement (JPCP) Thickness Design Methods Common in the United States, 12th International Symposium on Concrete Roads.

Skar, A., Klar, A., and Levenberg, E. 2019, Load-Independent Characterization of Plate Foundation Support Using High-Resolution Distributed Fiber-Optic Sensing, Sensors, 19(16), 3518.

Walker, W.W. and Holland, J.A. 2016, Modulus of Subgrade Reaction – Which One Should be Used?, Structural Service, Inc.

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