**Suggested Citation:**"Attachment B. Standard Method of Test for Determining the Cross-Anisotropic Resilient Modulus of Geosynthetic-Reinforced and Unreinforced Granular Material." National Academies of Sciences, Engineering, and Medicine. 2017.

*Quantifying the Influence of Geosynthetics on Pavement Performance*. Washington, DC: The National Academies Press. doi: 10.17226/24841.

**Suggested Citation:**"Attachment B. Standard Method of Test for Determining the Cross-Anisotropic Resilient Modulus of Geosynthetic-Reinforced and Unreinforced Granular Material." National Academies of Sciences, Engineering, and Medicine. 2017.

*Quantifying the Influence of Geosynthetics on Pavement Performance*. Washington, DC: The National Academies Press. doi: 10.17226/24841.

**Suggested Citation:**"Attachment B. Standard Method of Test for Determining the Cross-Anisotropic Resilient Modulus of Geosynthetic-Reinforced and Unreinforced Granular Material." National Academies of Sciences, Engineering, and Medicine. 2017.

*Quantifying the Influence of Geosynthetics on Pavement Performance*. Washington, DC: The National Academies Press. doi: 10.17226/24841.

**Suggested Citation:**"Attachment B. Standard Method of Test for Determining the Cross-Anisotropic Resilient Modulus of Geosynthetic-Reinforced and Unreinforced Granular Material." National Academies of Sciences, Engineering, and Medicine. 2017.

*Quantifying the Influence of Geosynthetics on Pavement Performance*. Washington, DC: The National Academies Press. doi: 10.17226/24841.

**Suggested Citation:**"Attachment B. Standard Method of Test for Determining the Cross-Anisotropic Resilient Modulus of Geosynthetic-Reinforced and Unreinforced Granular Material." National Academies of Sciences, Engineering, and Medicine. 2017.

*Quantifying the Influence of Geosynthetics on Pavement Performance*. Washington, DC: The National Academies Press. doi: 10.17226/24841.

**Suggested Citation:**"Attachment B. Standard Method of Test for Determining the Cross-Anisotropic Resilient Modulus of Geosynthetic-Reinforced and Unreinforced Granular Material." National Academies of Sciences, Engineering, and Medicine. 2017.

*Quantifying the Influence of Geosynthetics on Pavement Performance*. Washington, DC: The National Academies Press. doi: 10.17226/24841.

**Suggested Citation:**"Attachment B. Standard Method of Test for Determining the Cross-Anisotropic Resilient Modulus of Geosynthetic-Reinforced and Unreinforced Granular Material." National Academies of Sciences, Engineering, and Medicine. 2017.

*Quantifying the Influence of Geosynthetics on Pavement Performance*. Washington, DC: The National Academies Press. doi: 10.17226/24841.

**Suggested Citation:**"Attachment B. Standard Method of Test for Determining the Cross-Anisotropic Resilient Modulus of Geosynthetic-Reinforced and Unreinforced Granular Material." National Academies of Sciences, Engineering, and Medicine. 2017.

*Quantifying the Influence of Geosynthetics on Pavement Performance*. Washington, DC: The National Academies Press. doi: 10.17226/24841.

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157 ATTACHMENT B. STANDARD METHOD OF TEST FOR DETERMINING THE CROSS-ANISOTROPIC RESILIENT MODULUS OF GEOSYNTHETIC-REINFORCED AND UNREINFORCED GRANULAR MATERIAL

Standa Deter Geosy Mater AASHT 1. 1.1. 1.2. 1.3. 1.4. 2. 2.1. 3. 3.1. 3.2. 4. 4.1. rd Method mining nthetic ial O Design SCOP This te reinfor The va measur granula Cross-a calcula loads, a paveme This sta not pur the use determ REFE AASH ï§ T 3 SIGN The cro horizon perform The cro anisotr geosyn conditi APPA Rapid T confini resilien of Test f the Cro -Reinfo ation: T x E st method desc ced granular m lues of the cro e of the elastic r materials rec nisotropic res te the geosynth nd with pavem nt structures. ndard may in port to addres r of this standa ine the applica RENCED TO Standards: 07, Determini IFICANCE ss-anisotropic tal and vertica ance predictio ss-anisotropic opy nature of p thetic-reinforc ons (i.e., moist RATUS riaxial Test (R ng fluid during t modulus test or ss-Anis rced an x-xx ribed is applic aterials prepar ss-anisotropic modulus of th ognizing the s ilient modulus etic-reinforce ent design pro volve hazardou s all of the saf rd to consult a bility of regula DOCUMEN ng the Resilie AND USE resilient modu l resilient mod n of layered p resilient modu avement cons ed (i.e., geogri ure, density, e aTT) CellâT the test. A typ of granular m 158 otropic d Unrei able to unboun ed for testing resilient modu e unbound gra tress-dependen can be used w d or unreinforc cedures to de s materials, op ety concerns a nd establish a tory limitatio TS nt Modulus of lus test provid uli of paveme avement syste lus test provid truction mater d and geotexti tc.) and stress he pressure ce ical RaTT cel aterial is show Resilien nforced d granular ma by compaction lus determined nular material t nonlinear ch ith structural r ed pavement sign geosynthe erations, and ssociated with ppropriate saf ns prior to use Soils and Agg es a basic rela nt materials fo ms. es a means of ials, including le) granular m states. ll is used to co l suitable for u n in Figure 1. t Modu Granu terials and geo in the laborat from this pro s and geosynth aracteristics. esponse analy structural resp tic-reinforced equipment. Th its use. It is th ety and health . regate Materia tionship betw r the structura characterizing unbound gran aterials, under ntain the test s se in the cross lus of lar synthetic- ory. cedure are a etic-reinforce sis models to onse to wheel and unreinfor is standard d e responsibilit practices and ls een stress and l analysis and the cross- ular materials, a variety of pecimen and t -anisotropic d ced oes y of and he

4.2. 4.3. 4.4. 4.5. 5. 5.1. 5.2. 5.3. 5.4. 5.5. Figure Loadin electro cycles Confin system Specim of two two (or describ Specim prepare compac PREP Use 15 Cut the Prepare approx location Ensure than Â± Protect membr from st exceed Horizon Air 1âTypical Ra g DeviceâTh -pneumatic tes of haversine-sh ing Pressure Sy with a functio en Response M linear variable three) LVDTs ed in T 307. en Preparatio specimens of tion equipmen ARATION 0-mm-diamete geosynthetic p laboratory-co imate the in-si during comp that the moistu 0.5 percent for the prepared s ane and testing orage, weigh t s 0.5 percent, t tal LVDT Intake pid Triaxial Te e loading devic ting machine w aped load pul stemâThe co n generator tha easuring Equ deformation t on the lateral n Equipmentâ different mate t and compact OF TEST r and 150-mm roduct in a ci mpacted speci tu wet density action. re content of granular mate pecimens from within five d he specimen to hen do not tes 159 st Cell for Cro e shall be a to ith a function se with 1.5-sec nfining pressu t is capable of ipmentâThe m ransducers (LV direction of th Use of differe rials and to sim ion procedure SPECIMEN -height specim rcle with a diam mens/reconsti and moisture c the laboratory- rials from the moisture cha ays of complet determine if t t the prepared ss-Anisotropi p-loading, clo generator that ond loading a re shall be con applying hav easuring syst DTs) mounte e specimen. T nt methods of ulate desired s shall be as de S ens for tests o eter of 150 m tuted test spec ontent. Place compacted sp in-situ moistur nge by applyin ion. Prior to st here was any specimen. c Resilient Mo sed-loop, elect is capable of nd 1.5-second trolled by an ersine-shaped em for all mat d on the top of he LVDT requ compaction is field condition scribed in T 3 n granular ma m. imens of granu the geosynthet ecimen does n e content obta g the triaxial orage, and dir moisture loss. Ra Co Se Elec Cont dulus Test ro-hydraulic, o applying repea unloading per electro-pneum confining pres erials shall con the specimen irement shall necessary to s. The specim 07. terial specimen lar materials t ic at a certain ot vary by mor ined. cell or the triax ectly after rem If moisture los TT Cell nfining Pressu nsor tro-hydraulic rol System r ted iods. atic sure. sist and be as en s. o e ial oval s re

6. 6.1. 6.2. 6.3. 6.4. Stress Sequence 0* 1 2 3 4 5 6 7 8 9 10 *Note: Stre PROC Place th cell as Connec pressur maxim Begin t 103.4 k precon to elim Apply stress s pressur Table 1âC Static Stres Axial 206.8 40 50 70 130 150 170 220 250 250 250 ss Level 0 is f EDURE e laboratory-c shown in Figu t the air pressu e of 41.4 kPa t um applied axi he test by appl Pa and 103.4 k ditioning step t inate the initia a specified axi equence and lo e to the lateral Figure 2âC ross-Anisotrop s (kPa) Lateral A 103.4 25 25 40 60 70 100 120 140 120 105 or preconditio ompacted geo re 2. re line to the o the test spec al stress durin ying 500 repe Pa confining o eliminate th l loading versu al load to the t ading mode sh direction of th onfiguration o ic Resilient M Unreinforce Compression xial Later 5 0 10 0 10 0 20 0 20 0 20 0 30 0 30 0 30 0 30 0 ning of granul 160 synthetic-reinf RaTT cell and imen. Maintai g each stress le titions of a loa pressure accor e effects of the s reloading. op of the triaxi own in Table e test specime f Cross-Aniso odulus Testing d Granular Ma Dynamic S Sh al Axial 10 10 10 20 20 20 30 30 30 30 ar material. orced or unrei apply the spec n a contact stre vel. d equivalent to ding to Stress interval betw al cell piston r 1. Meanwhile n according to tropic Resilien Protocol for terial tress (kPa) ear Lateral â5 â5 â5 â10 â10 â10 â15 â15 â15 â15 nforced specim ified precondi ss of 10 perce a maximum a Level 0 in Tab een compactio od for 100 cyc , apply a speci Table 1. t Modulus Tes Geosynthetic-R Extension Axial Late â5 5 â10 5 â10 10 â10 10 â10 10 â20 20 â20 20 â20 20 â20 20 â20 20 Un Te M R Te Ag Sp ens in the Ra tioning confin nt Â± 0.7 kPa o xial stress of le 1. This is a n and loading les according fied confining t einforced and No. of L Applicaral 500 100 100 100 100 100 100 100 100 100 100 iversal sting achine apid Triaxial st Cell gregate ecimen TT ing f the and to oad tions

161 7. CALCULATION OF RESULTS 7.1. Determine the cross-anisotropic resilient modulus shown in Equation 1 using the system identification method (see Annex A). 1 1 xy xx x x x x x y yxy xy x x y x E E E E E E ïµ ïµ ï³ ï¥ï³ ï¥ïµ ïµ ï³ ï ï ï½ ï ï ï© ï¹ ï¬ ï¼ïª ïº ï¬ ï¼ï¯ ï¯ïª ïº ï ï½ ï ï½ïª ïº ï® ï¾ï¯ ï¯ïª ïº ï® ï¾ï« ï» (1) where: xï³ = the stress in the lateral direction; yï³ = the stress in the axial direction; xï¥ = the resilient strain in the lateral direction; yï¥ = the resilient strain in the axial direction; xE = the resilient modulus in the lateral direction; yE = the resilient modulus in the axial direction; xxï® = the Poissonâs ratio in the lateral plane; and xyï® = the Poissonâs ratio in the axial plane. 8. REPORT 8.1. The report shall include the following: 8.1.1. Source of granular material. 8.1.2. Type of geosynthetic. 8.1.3. Geosynthetic location. 8.1.4. Determined cross-anisotropic resilient modulus for the tested stress levels.

162 Annex AâSystem Identification Method The objective of the system identification method is to estimate the system characteristics using only input and output data from the system to be identified. The model is identified when the error between the model and the real process is minimized in some sense; otherwise, the model must be modified until the desired level of agreement is achieved. The system identification method requires the accurately measured output data of the unknown system, a suitable model to represent the behavior of the system, and an efficient parameter adjustment algorithm that converges accurately and rapidly. An algorithm can be developed for adjusting model parameters on the basis of Taylorâs series expansion. Let the mathematical model of some process be defined by n parameters: 1 2( , ,..., ; , )nf f p p p x tï½ (A-1) where x and t are independent spatial and temporary variables. Then any function 1 2( , ,..., ; , )k n k kf p p p x t can be expanded in a Taylorâs series as: 2( ) ( )k k kf p p f p f p ï¤ï« ï ï½ ï«ï ïï ï« (A-2) Assuming ( )kf p pï«ï to be the actual output of the system and ( )kf p to be the output of the model for the most recent set of parameters, the error between the two outputs becomes: 1 2 1 2 ( ) ( )k k k k k k k n n e f p p f p f p f f fp p p p p p ï¼ï¯ï½ ï« ï ï ï¯ï¯ï½ ï ïï ï½ï¯ï¶ ï¶ ï¶ ï¯ï½ ï ï« ï ï« ïïï ï« ïï¶ ï¶ ï¶ ï¯ï¾ (A-3) where ke represents the difference between the actual system output and the model output for each observed point k . If the error is evaluated at m values ( m nï³ ) of the independent variables, m equations will be generated as: 1 1 i n k m k k i i ik i k i k e f p p f p f p ï½ï½ ï½ï½ ï¶ ïï½ ï¶ï¥ (A-4) This can be collected into a matrix form as: r Fï¡ï½ (A-5)

163 where: k k k er f ï½ (A-6) k i ki i k f pF p f ï¶ï½ ï¶ (A-7) i i p p ï¡ ïï½ (A-8) The r vector is completely determined from the outputs of the model and the real system. The matrix F is called the sensitivity matrix, and its elements kiF reflect the sensitivity of the output kf to the parameter ip . It is generated by the differentials of the output kf with respect to the parameter ip . The unknown vector ï¡ reflects the relative changes of the parameters. Once the vector ï¡ is obtained, a new set of parameters is determined as: 1 (1 )r ri ip p ï¡ï« ï½ ï« (A-9) where r is the iteration number. The iteration process is continued until the desired convergence is reached. Applying the algorithm described above to the rapid triaxial testing model results in four unknown parameters ( , , ,y x xy xxE E ïµ ïµ ) and two outputs ( ,x yï¥ ï¥ï ï ). The model strains (model output) can be determined from the values of the parameters, which can be guessed initially from the system output. The difference between the measured strains and the model strains (model output) represents the error, which can be improved through the parameter adjustment routine until a desired criterion is achieved. The F matrix, ï¡ vector, and r vector can be derived from Equation A-4 as follows: ^ ^ ^ ^ ^ ^ ^ ^ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) xy yx x x x xx x x x xy x xx x y x y y xy y y yx xx x y xy y xx y y y EE E E F EE E E ïµï¥ ï¥ ï¥ ïµ ï¥ ï¥ ïµ ï¥ ïµ ï¥ ï¥ ï¥ ï¥ ïµ ï¥ ï¥ïµ ï¥ ïµ ï¥ ïµ ï¥ ï¥ ï© ï¹ï¶ ï ï¶ ï ï¶ ï ï¶ ïï ï ï ïïª ïºï¶ ï ï¶ ï ï¶ ï ï¶ ïïª ïºï½ ïª ïºï¶ ï ï¶ ï ï¶ ï ï¶ ïïª ïºï ï ï ïï¶ ï ï¶ ï ï¶ ï ï¶ ïïª ïºï« ï» (A-10) 1 11 1 Tr rr r xy yx xx r r r r x xy xx y EE E E ïµ ïµï¡ ïµ ïµ ï« ï«ï« ï«ï© ï¹ï½ ïª ïºïª ïºï« ï» (A-11)

164 ^ ^ ^ ^ m x x x m y y y r ï¥ ï¥ ï¥ ï¥ ï¥ ï¥ ï© ï¹ï ïïïª ïºïïª ïºï½ ïª ïºï ïïïª ïºïïª ïºï« ï» (A-12) where: m xï¥ï = measured (actual system) radial strain, m yï¥ï = measured (actual system) axial strain, ^ xï¥ï = calculated (model) radial strain, and ^ yï¥ï = calculated (model) axial strain. To generate enough elements in the sensitivity matrix and to control the number of row degeneracy, the three stress regimes (triaxial compression, triaxial shear, and triaxial extension) are combined to give one F matrix and one r matrix at each stress state. Thus, at each stress state, they can be written as: TC TC TS TS TE TE F r F r F r ï¡ ï© ï¹ ï© ï¹ï¬ ï¼ïª ïº ïª ïºï¯ ï¯ ï½ï ï½ïª ïº ïª ïºï¯ ï¯ïª ïº ïª ïºï® ï¾ï« ï» ï« ï» (A-13) where: TCF = sensitivity matrix for triaxial compression regime, TSF = sensitivity matrix for triaxial shear regime, TEF = sensitivity matrix for triaxial extension regime, TCr = r vector for triaxial compression regime, TSr = r vector for triaxial shear regime, and TEr = r vector for triaxial extension regime. Four of the five material properties ( , , ,y x xy xxE E ïµ ïµ ) can be determined by the above algorithm.