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

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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.