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Proposed Enhancements to Pavement ME Design: Improved Consideration of the Influence of Subgrade and Unbound Layers on Pavement Performance (2019)

Chapter: Appendix B. Definitions of Model Parameters of Unbound Layer and Subgrade Models

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Suggested Citation:"Appendix B. Definitions of Model Parameters of Unbound Layer and Subgrade Models." National Academies of Sciences, Engineering, and Medicine. 2019. Proposed Enhancements to Pavement ME Design: Improved Consideration of the Influence of Subgrade and Unbound Layers on Pavement Performance. Washington, DC: The National Academies Press. doi: 10.17226/25583.
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Suggested Citation:"Appendix B. Definitions of Model Parameters of Unbound Layer and Subgrade Models." National Academies of Sciences, Engineering, and Medicine. 2019. Proposed Enhancements to Pavement ME Design: Improved Consideration of the Influence of Subgrade and Unbound Layers on Pavement Performance. Washington, DC: The National Academies Press. doi: 10.17226/25583.
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Suggested Citation:"Appendix B. Definitions of Model Parameters of Unbound Layer and Subgrade Models." National Academies of Sciences, Engineering, and Medicine. 2019. Proposed Enhancements to Pavement ME Design: Improved Consideration of the Influence of Subgrade and Unbound Layers on Pavement Performance. Washington, DC: The National Academies Press. doi: 10.17226/25583.
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Suggested Citation:"Appendix B. Definitions of Model Parameters of Unbound Layer and Subgrade Models." National Academies of Sciences, Engineering, and Medicine. 2019. Proposed Enhancements to Pavement ME Design: Improved Consideration of the Influence of Subgrade and Unbound Layers on Pavement Performance. Washington, DC: The National Academies Press. doi: 10.17226/25583.
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Suggested Citation:"Appendix B. Definitions of Model Parameters of Unbound Layer and Subgrade Models." National Academies of Sciences, Engineering, and Medicine. 2019. Proposed Enhancements to Pavement ME Design: Improved Consideration of the Influence of Subgrade and Unbound Layers on Pavement Performance. Washington, DC: The National Academies Press. doi: 10.17226/25583.
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Suggested Citation:"Appendix B. Definitions of Model Parameters of Unbound Layer and Subgrade Models." National Academies of Sciences, Engineering, and Medicine. 2019. Proposed Enhancements to Pavement ME Design: Improved Consideration of the Influence of Subgrade and Unbound Layers on Pavement Performance. Washington, DC: The National Academies Press. doi: 10.17226/25583.
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Suggested Citation:"Appendix B. Definitions of Model Parameters of Unbound Layer and Subgrade Models." National Academies of Sciences, Engineering, and Medicine. 2019. Proposed Enhancements to Pavement ME Design: Improved Consideration of the Influence of Subgrade and Unbound Layers on Pavement Performance. Washington, DC: The National Academies Press. doi: 10.17226/25583.
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Suggested Citation:"Appendix B. Definitions of Model Parameters of Unbound Layer and Subgrade Models." National Academies of Sciences, Engineering, and Medicine. 2019. Proposed Enhancements to Pavement ME Design: Improved Consideration of the Influence of Subgrade and Unbound Layers on Pavement Performance. Washington, DC: The National Academies Press. doi: 10.17226/25583.
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Suggested Citation:"Appendix B. Definitions of Model Parameters of Unbound Layer and Subgrade Models." National Academies of Sciences, Engineering, and Medicine. 2019. Proposed Enhancements to Pavement ME Design: Improved Consideration of the Influence of Subgrade and Unbound Layers on Pavement Performance. Washington, DC: The National Academies Press. doi: 10.17226/25583.
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Suggested Citation:"Appendix B. Definitions of Model Parameters of Unbound Layer and Subgrade Models." National Academies of Sciences, Engineering, and Medicine. 2019. Proposed Enhancements to Pavement ME Design: Improved Consideration of the Influence of Subgrade and Unbound Layers on Pavement Performance. Washington, DC: The National Academies Press. doi: 10.17226/25583.
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Suggested Citation:"Appendix B. Definitions of Model Parameters of Unbound Layer and Subgrade Models." National Academies of Sciences, Engineering, and Medicine. 2019. Proposed Enhancements to Pavement ME Design: Improved Consideration of the Influence of Subgrade and Unbound Layers on Pavement Performance. Washington, DC: The National Academies Press. doi: 10.17226/25583.
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B-1 Appendix B. Definitions of Model Parameters of Unbound Layer and Subgrade Models MODULUS MODELS OF UNBOUND LAYERS AND SUBGRADE 2 1 3 k RM k  where RM is the resilient modulus, 3 is the confining pressure, and 1k and 2k are regression coefficients. 2 1 k RM k  where RM is the resilient modulus,  is the bulk stress, and 1k and 2k are regression coefficients.  2 3 1R dM k k k    1 dk   2 4 1R dM k k k   1 dk  where RM is the resilient modulus; d is the deviatoric shear stress; and 1k , 2k , 3k , and 4k are regression coefficients. ' ' d R d a bM     where RM is the resilient modulus, d is the deviatoric shear stress, and 'a and 'b are regression coefficients. 32 1 kk R dM k   where RM is the resilient modulus; d is the deviatoric shear stress;  is the bulk stress; and 1k , 2k , and 3k are regression coefficients. 2 3 1 1 k k oct R IM k Pa Pa Pa             where RM is the resilient modulus; 1I is the first invariant of stress tensor; oct is the octahedral shear stress; Pa is the atmospheric pressure; and 1k , 2k , and 3k are regression coefficients. 2 3 1 1 1 k k oct R IM k Pa Pa Pa             where RM is the resilient modulus; 1I is the first invariant of stress tensor; oct is the octahedral shear stress; Pa is the atmospheric pressure; and 1k , 2k , and 3k are regression coefficients.

B-2 2 1 2 R I JM M Pa R Pa Pa             where RM is the resilient modulus; 1I is the first invariant of stress tensor; 2J is the second invariant of shear stress tensor; Pa is the atmospheric pressure; R is a function of the Poisson’s ratio; and M and  are model coefficients.   log 1 exp ln R Ropt m opt M b aa bM k S S a          where RM is the resilient modulus at a given degree of saturation; RoptM is the resilient modulus at reference condition; a is the minimum of log R Ropt M M        ; b is the maximum of log R Ropt M M        ; mk is the regression parameter; and  optS S is the variation of degree of saturation expressed in decimal.    2 3 1R d s a wM k k k k u u        2 4 1R d s a wM k k k k u u     where RM is the resilient modulus; d is the deviatoric shear stress; au is the air pressure; wu is the pore water pressure; and 1k , 2k , 3k , 4k , and sk are regression coefficients. 2 3 1 4 1 3 k koct R I kM k Pa Pa Pa             where RM is the resilient modulus; 1I is the first invariant of stress tensor; oct is the octahedral shear stress; Pa is the atmospheric pressure; and 1k , 2k , 3k , and 4k are regression coefficients. 2 3 1 1 3 k km oct R I fhM k Pa Pa Pa              where RM is the resilient modulus; 1I is the first invariant of stress tensor; oct is the octahedral shear stress; Pa is the atmospheric pressure;  is the volumetric water content; f is the saturation factor; mh is the matric suction; and 1k , 2k , and 3k are regression coefficients.

B-3 2 3 1 1 1 3 3 k km oct oct R II f h M k Pa Pa Pa                        where RM is the resilient modulus; 1I is the first invariant of stress tensor; oct is the octahedral shear stress; Pa is the atmospheric pressure;  is the volumetric water content; f is the saturation factor; mh is the matric suction;  and  are the Henkel pore water pressure coefficients; and 1k , 2k , and 3k are regression coefficients.   21 k R d w mM k     where RM is the resilient modulus; d is the deviatoric shear stress; w is the Bishop’s effective stress coefficient; m is the matric suction; and 1k and 2k are regression coefficients. 2 3 1 1 k k w m oct R a a a M k P P P                 where RM is the resilient modulus;  is the bulk stress; oct is the octahedral shear stress; Pa is the atmospheric pressure; w is the Bishop’s effective stress coefficient; m is the matric suction; and 1k , 2k , and 3k are regression coefficients. ' 2 4 4 ' 1 3 1 1o k k k m mnet w sat oct R a a a a uM k P P P P                         where RM is the resilient modulus; net is the net bulk stress; oct is the octahedral shear stress; Pa is the atmospheric pressure; w satu  is the build-up of pore water pressure under saturated conditions; 0m is the initial matric soil suction; m is the relative change of matrix soil suction with respect to 0m ; and ' 1k , ' 2k , ' 3k , and ' 4k are regression coefficients.   2 3 6 1 7 3 k k b oct R a us a a w a a kM k p k k p p p                     where RM is the resilient modulus; b is the bulk stress; oct is the octahedral shear stress; Pa is the atmospheric pressure; au is the air pressure; wu is the pore water pressure;  is the normalized volumetric water content;  is the fitting parameter; and 1k , 2k , 3k , 6k , 7k , and usk are regression coefficients.

B-4 2 3 4 1 3 1 k k w oct R a a a k SVM k P P P               where RM is the resilient modulus;  is the bulk stress; oct is the octahedral shear stress; Pa is the atmospheric pressure; wV is the volumetric water content; S is the soil suction; and 1k , 2k , 3k , and 4k are regression coefficients. 2 3 1 1 1 ; k k V oct R a a a H R VH V V R R IM k P P P M Gs r M M                where VRM is the resilient modulus in the vertical direction; 1I is the first invariant of stress tensor; oct is the octahedral shear stress; Pa is the atmospheric pressure; 1k , 2k , and 3k are regression coefficients; HRM is the resilient modulus in the horizontal direction; VHG is the shear modulus in the horizontal-vertical plane; and s and r are the modulus ratios. 2 3 5 6 8 9 1 4 7 ; ; k k k k V Hoct oct R a R a a a a a k k oct VH a a a M k P M k P P P P P G k P P P                                             where VRM is the resilient modulus in the vertical direction; oct is the octahedral shear stress; Pa is the atmospheric pressure; HRM is the resilient modulus in the horizontal direction; VHG is the shear modulus in the horizontal-vertical plane; and 1k , 2k , 3k , 4k , 5k , 6k , 7k , 8k , and 9k are regression coefficients. 2 3 1 1 3 ; k k V m oct R a a a H R VH V V R R I fhM k P P P M Gn m M M                 where VRM is the resilient modulus in the vertical direction; 1I is the first invariant of stress tensor; oct is the octahedral shear stress; Pa is the atmospheric pressure;  is the volumetric water content; f is the saturation factor; mh is the matric suction; 1k , 2k , and 3k are regression coefficients; HRM is the resilient modulus in the horizontal direction; VHG is the shear modulus in the horizontal-vertical plane; and n and m are the modulus ratios.

B-5       1 2 3 1.3577 0.0106 % 0.0437 0.5193 0.0073 4 0.0095 40 0.0027 200 0.003 0.0049 1.4258 0.0288 4 0.0303 40 0.0521 200 0.0251 % 0.0535 0.0672 0.0026 0.0025 0.6055 k clay wc k P P P LL wopt k P P P silt wcLL wopt opt s wopt                     where 1k , 2k , and 3k are resilient modulus model coefficients; %clay is the clay content in percentage; % silt is the silt content in percentage; 4P is the percent of material passing sieve No. 4; 40P is the percent of material passing sieve No. 40; 200P is the percent of material passing sieve No. 200; LL is the liquid limit; wopt is the optimum water content; wc is the water content; and opt is the dry density at optimum water content.               1 2 3 ln 137.19 13.60ln 4.35ln 0.62ln 36.14 0.04 3.81ln 0.22 0.77ln 4.39 0.45ln 0.01 0.05 0.15ln d A T A s T d s T k k pfc a k pfc a                        where 1k , 2k , and 3k are resilient modulus model coefficients; d is the dry density; A is the scale factor of angularity index; T is the scale factor of texture index; pfc is the percent fines content; and sa is the shape factor of angularity index.  0.642555rM CBR where rM is the resilient modulus, and CBR is the California bearing ratio. 1155 555rM R  where rM is the resilient modulus, and R is the resistance R-value. 30000 0.14 i r aM       where rM is the resilient modulus, and ia is the AASHTO layer coefficient.   0.64 752555 1 0.728r M wPI        where RM is the resilient modulus, wPI is the weighted plasticity index. 0.64 1.12 2922555rM DCP       where RM is the resilient modulus, and DCP is the dynamic cone penetrometer index.

B-6 PERMANENT DEFORMATION MODELS OF UNBOUND LAYERS AND SUBGRADE  1 P r N N N         where P is the accumulated plastic strain; r is the resilient strain of granular material; N is the number of load cycles; and  and  are regression coefficients. p b r aN    where p is the accumulated plastic strain; r is the resilient strain of granular material; N is the number of load cycles; and a and b are regression coefficients. 0 N p e         where p is the accumulated plastic strain; N is the number of load cycles; and 0 ,  , and  are regression coefficients.    , 0 kzp p zz e   where  p z is the plastic strain at depth z; , 0p z  is the vertical plastic strain at the top of subgrade; z is the depth measured from the top of subgrade; and k is the model coefficient. 0 N p s v r e               where p is the accumulated plastic strain; r is the resilient strain of granular material; N is the number of load cycles; 0 ,  , and  are regression coefficients; s is a global calibration coefficient, 1.673 for granular materials; r is the resilient strain imposed in the laboratory test; and v is the average vertical resilient strain in the base layer of the flexible pavements. 6 6 0 1 2 0 1 2log .log p oct oct r k ka a a b b b N Pa Pa Pa Pa                                              where P is the accumulated plastic strain; r is the resilient strain of granular material; N is the number of load cycles;  is the bulk stress; Pa is the atmospheric pressure; oct is the octahedral shear stress; and 0a , 1a , 2a , 0b , 1b , and 2b are model coefficients.

B-7 1 b p RCN R    ' f qb d c q         where P is the accumulated plastic strain; N is the number of load cycles; C is the permanent strain in the first loading cycle; b is a shear ratio parameter; R is the shear failure ratio 1 3 0f q q q Mp      , 6sin 3 sin M     , 0 6cos 3 sin cq      , where c and  are cohesion and friction angle; and d and 'c are material parameters.  max D fB C p dAN            where P is the accumulated plastic strain; N is the number of load cycles; d is the deviatoric shear stress; f is shear stress; m ax is shear strength; and A , B , C , and D are regression coefficients.    0 2 1m nNp e J I K             2sin 3 3 sin       6cos 3 3 sin cK      where P is the accumulated plastic strain; N is the number of load cycles; and 0 ,  , and  are regression coefficients; 2J is the second invariant of the deviatoric stress tensor; 1I is the first invariant of the stress tensor; 0 ,  ,  , m and n are model coefficients; and c and  are cohesion and friction angle, respectively. 60 6 4 2 3 2 6 log 0.80978 0.06626 0.003077 10 log 0.9190 0.03105 0.001806 1.5 10 log 1.78667 1.45062 3.784 10 2.074 10 1.05 10 c r r c r c c r W E W E W W E                                          where 0 ,  , and  are Pavement ME Design model coefficients; r is the resilient strain of granular material; N is the number of load cycles; cW is the water content;  is the bulk stress; and rE is the resilient modulus of granular material.

B-8     910 0 1 9 9 0.15 20 2 log 0.61119 0.017638 4.8928510 1 10 r c e e W                                         where 0 ,  , and  are Pavement ME Design model coefficients; r is the resilient strain of granular material; and cW is the water content. 0ln 10.24 0.03 0.10 0.88 3.95 ln ln 6.74 0.02 0.04 0.85 0.03 0.13 ln 10.17 2.75 ln 0.05 2.00 1.61ln 0.34 A T G G T d G A T MBV pfc a MBV pfc a a pfc a a                         where 0 ,  , and  are Pavement ME Design model coefficients; MBV is methylene blue value; pfc is the percent fines content; T is the scale factor of texture index; Aa is the shape factor of angularity index; A is the scale factor of angularity index; d is the dry density; Ga is the shape factor of gradation; Ta is the shape factor of texture index; and G is the scale factor of gradation. SHEAR STRENGTH MODELS OF UNBOUND LAYERS AND SUBGRADE tannc    where  is the shear stress; n is the normal stress; c is the cohesion; and  is the friction angle.    ' tan ' tan bf n a a wc              where f is the shear strength; n is the normal stress; 'c is the cohesion; ' is the effective angle of shearing resistance for a saturated soil; b is the angle of shearing resistance with respect to matric suction; au is the air pressure; wu is the pore water pressure; and  is the fitting coefficient.   tantan nba wc u u     where  is the shear stress; n is the normal stress; au is the air pressure; wu is the pore water pressure; 'c is the cohesion;  is the friction angle for a saturated soil; and b is the angle of shearing resistance with respect to matric suction.

B-9    ' tan ' tan bf n a a wc S           where f is the shear strength; n is the normal stress; 'c is the cohesion; ' is the effective angle of shearing resistance for a saturated soil; b is the angle of shearing resistance with respect to matric suction; au is the air pressure; wu is the pore water pressure; and S is the degree of saturation.    ' tan ' tan bf n a a wc             where f is the shear strength; n is the normal stress; 'c is the cohesion; ' is the effective angle of shearing resistance for a saturated soil; b is the angle of shearing resistance with respect to matric suction; au is the air pressure; wu is the pore water pressure;  is the normalized volumetric water content; and  is the fitting parameter.        ' tan ' tanf n a a w a w a w bc                        where f is the shear strength; n is the normal stress; 'c is the cohesion; ' is the effective angle of shearing resistance for a saturated soil;  is the angle of shearing resistance with respect to matric suction; au is the air pressure; wu is the pore water pressure; and  and  are model coefficients.  ' tan ' ' 1676.624 2.088 13.260 0.113 270.722ln 38.778 ' 2.827 0.016 0.0005 0.051 0.763ln 0.008 n m A A d G A S d c c MBV a fh a MBV a pfc                          where  is the shear stress; n is the normal stress; 'c is the cohesion; ' is the friction angle;  is the volumetric water content; f is the saturation factor; mh is the matric suction; MBV is the methylene blue value; Aa is the shape factor of angularity index; A is the scale factor of angularity index; d is the dry density; Ga is the shape factor of gradation; Sa is the shape factor of shape index; and pfc is the percent fines content. 2 tan 83.95 1.58 40 2.57 0.043 40 6.88 0.14 0.81 tan 1.61 0.96 0.88 4.13 31.82 n N N sN sb c c N n N PL G PI n G                       where 40N is the percent of material passing 0.42 mm sieve size; n is the porosity; 40NN is normalized 40N =  40 55.89N  ; NPL is normalized plastic limit = 15.89PL  ; sNG is normalized specific gravity of aggregate = 2.61sG  ;  is matric suction; PI is plasticity index; and sbG is specific gravity of binder content.

B-10 EROSION MODELS OF UNBOUND LAYERS ESALg          where g is the amount of distress as a fraction of a pumping level of 3; ESAL is the equivalent 80 kN single axle loads; and  is the model coefficient. log 1.07 0.34 i dP m ESAL f m D       where iP is the pumping index; ESAL is the equivalent 80 kN single axle loads; df is the drainage adjustment factor; m is the model coefficient; and D is the slab thickness. exp 2.884 1.652 log 10,000 ESAL DE NPI               where NPI is the normalized pumping index of volume of pumped material; ESAL is the equivalent 80 kN single axle loads; and DE is the deformation energy per one application of ESAL. 36.67 2.884 1.652 log 10,000 P NPI ESAL DE NPI F                  where P is the volume of pumped material; NPI is the normalized pumping index of volume of pumped material; ESAL is the equivalent 80 kN single axle loads; and DE is the deformation energy per one application of ESAL.  0.1031 2 1 log 14.524 6.777 9.0 100 m i i i N C P C nPercent erosion damage N      where N is the allowable number of load repetitions based on a pressure of a PSI of 3.0; 1C is the adjustment factor; P is the pressure on the foundation under the slab corner; m is the total number of load groups; 2C is the model coefficient; in is the predicted number of repetitions for the ith load group; and iN is the allowable number of repetitions for the ith load group.    0% D Nf Erosion f e          where 0f is maximum faulting;  %f Erosion is percent of maximum faulting;  is scale calculation factor based on laboratory erosion test;  D N is damage after N load repetitions;

B-11 v is time delay before the appearance of visible (measurable) damage; and  is shape factor related to the erosion rate. FOUNDATION MODELS OF SUBGRADE (x,y) kw(x,y)p  where  ,p x y is the distributed load applied in the x-y plane;  ,w x y is the displacement in the vertical direction; and k is the foundation modulus. 2(x, y) kw(x, y) (x, y)p T w   where  ,p x y is the distributed load applied in the x-y plane; w is the displacement in the vertical direction; 2 is the Laplace operator in x and y ( 2 2 2 2 2x y          ); and T is the constant tension of a stretched elastic membrane of the top ends of the springs. 2 2(x, y) kw(x, y) (x, y)p D w    where  ,p x y is the distributed load applied in the x-y plane; w is the displacement in the vertical direction; 2 is the Laplace operator in x and y ( 2 2 2 2 2x y          ); and D is the flexural rigidity of the plate. 2(x, y) kw(x, y) G (x, y)p w   where  ,p x y is the distributed load applied in the x-y plane, w is the displacement in the vertical direction; 2 is the Laplace operator in x and y; and G is the constant of the shear layer. 2 2(1 ) p kw G c 3k 4 3 4 9 f f k G p w c c E k H G HG          where p is the distributed load; w is the displacement in the vertical direction; 2 is the Laplace operator in x and y ( 2 2 2 2 2x y          ); c is the spring constant of upper spring layer; k is the spring constant of lower spring layer; G is the constant of the shear layer; fE is Young’s modulus; fG is the shear modulus of the foundation material; and H is the thickness of the foundation.

Next: Appendix C. Evaluation and Screening of Unbound Layer and Subgrade Models »
Proposed Enhancements to Pavement ME Design: Improved Consideration of the Influence of Subgrade and Unbound Layers on Pavement Performance Get This Book
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The performance of flexible and rigid pavements is known to be closely related to properties of the base, subbase, and/or subgrade. However, some recent research studies indicate that the performance predicted by this methodology shows a low sensitivity to the properties of underlying layers and does not always reflect the extent of the anticipated effect, so the procedures contained in the American Association of State Highway and Transportation Officials’ (AASHTO’s) design guidance need to be evaluated.

NCHRP Web-Only Document 264: Proposed Enhancements to Pavement ME Design: Improved Consideration of the Influence of Subgrade and Unbound Layers on Pavement Performance proposes and develops enhancements to AASHTO's Pavement ME Design procedures for both flexible and rigid pavements, which will better reflect the influence of subgrade and unbound layers (properties and thicknesses) on the pavement performance.

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