Modeling of Granular Materials in Pavements

Transcription

1 Transportation Researh Reord Modeling of Granular Materials in Pavements S. F. BROWN and J. W. PAPPIN ABSTRACT The problem of theoretial modeling of granular materials in pavements is onsidered i a previously published tehnique and assoiated materials data are used. A detailed stress-resilient strain model was used in a finite element onfiguration that is based on a seant modulus approah. A parametri theoretial study involving 56 different pavement strutures with two granular materials provided extensive data on the in situ stress onditions in unbound layers and their equivalent stiffnesses. The inidene of failure elements is disussed and the onlusion is drawn that the simple T<- nonlinear model and linear elasti layered systems are inadequate for omputing stresses within the granular layer. Arbitrary adjustments to omputed stresses that indiate apparent failure or tensile onditions are unneessary when an aurate material model and assoiated omputational tehniques are used. The onept of a fixed modular ratio between a granular layer and a subgrade was found to be inappropriate beause a partiular granular material has an essentially onstant equivalent stiffness. Linear elasti layered system omputer programs an be used to determine ritial design parameters when the granular layer stiffness is hosen on the basis of results from detailed nonlinear analysis. In the design of new roads and in the expanding field of pavement strutural evaluation, there is a ontinuing need for an adequate means of modeling unbound granular layers. The problem is not new; the nonlinear elasti properties of granular materials have long been appreiated and a number of tehniques have been used to take these into aount in strutural analysis. These methods have inluded an iterative approah us.ing linear elasti layered systems, first outlined by Monismith et al. (1) and appliat'lon of the finite element method ( 2)-: This latter tehnique has been used as a basis for developing nomographi proedures for pavement design (3). The major.tty of the work done in this field har used the so-alled K-6 model to desdbe the nonlinear elasti harateristis of granular materials. This model was developed 1'rom repeated load triaxial test results and is of the form: where e resilient modulus, whih is the repeated deviator stress divided by the axial resilient strain: peak value of the sum of the prinipal stresses: and material onstants. Brown and Pappin (_1) have desribed a more detailed model for granular materials, whih has wider appliability, and have disussed the limitations of the K- model. They also presen ted a omputational proedure to inorporate their model i n a finite element pakage, known as SENOLr to analyze pavement strutures. The use of their proedure has also been illustrated (). The SENOL omputer program has sine been used to analyze a wide range of pavements and the results have thrown some additional 1 ight on the in situ behavior of granular materials. use of the K-& model has also been further investigated to establish its limitations. Beause finite element analysis is still regarded as essentially a researh tool (1) in pavement engineering, SENOL has also been used to alibrate simpler analysis tehniques based on linear elasti layered systems. The limitations of these have also been established. GRANULAR MATERIAL MODELS The resilient strain model desribed by Brown and Pappin C!,. _) was developed from a omprehensive set of repeated load triaxial test data. The strains were expressed i n terms of resilient shear and volumetri omponents leading to stress-dependent shear and bulk modu.li. Stresses were expressed in terms of the invariants, mean no rmal effetive stress (p' = &/3) and deviator stress (q). This model is referred to as the "" beause it is best illustrated, as in Figure 1, by use of strain ontours in p'-q stress spae. Two materials are onsidered in this paper. The first, Model A, is a well-graded rushed limestone, and the seond, Model B, is a uniformly graded material from the same soure. They were seleted to represent good- and poor-quality material in terms of stiffness. Details of both models have been presented by Brown and Pappin (4). Coeffiients in the 'Orresponding T<- relationships for these two materials are as follows: Model A: K kpa, K2 =,69 Model B: K 1 = kpa, The ontour models annot be expressed as suintly as this, so referene should be made to Brown and Pappin <!i> for full details. Figure 2 shows a typial stress pulse ln p'-q spae for an element of granular material in a pavement. Point A represents overbuz:den pressure and Point B is the peak stress that ours when the wheel load is i.ed lately above the element. The ontour model was developed from a large numbe r of stress paths suh as AB overing the stress spae of Figure 2 but limited to peak values of q/p = This was done to avoid the development of signifiant permanent

2 46 Transportation Researh Reord 122 'm ;: Mean normal effetive stress (p') The K-6 model was developed from tests involving onstant onfining stress and deviator stress pulsed from zero to a peak value. The stress paths in Figure 2 for suh tests would involve Point A being on the p' axis (equal to the onfining stress at that point) and the slope of AB being 3. The parameter B in Equation 1 is the value orresponding to Point B in Figure 2. Hene the ontour model is a more exat representation of the material behavior and is better able to predit stress onditions in a pavement in whih a wide range of stress paths is possible. In the omputations that were performed during this investigation, both the asphalt and the subgr<ide layers in eah pavement were assumed to be linearly elasti. When linear elasti! ty i s applied to materials the basi harateristis of whih a e eithe r nonlinear or visoelasti, the term "elasti stiffness" is used in plae of Young ' s modulus in this paper. p' Inreasing FIGURE 1 Contour model in p'-q stress spae. CD ;; - > Mean normal stress (p) FIGURE 2 Typial in situ stress path due to wheel loading. strains that our when peak stresses probe lose to the failure line; q/p = 2.2 for this material (both well graded and single sized). Some tests involving these high stress ratios showed that the basi resilient strain model was apable of extrapolation into this zone. COMPUTATIONAL PROCEDURES The finite element program SENOL was designed speifially to apply the ontour model for granular materials to pavement ana.lysis. Details of the omputational proedure have been published by Brown and Pappin (4,5) but will be s uarized here. The starting point for analysis in a partiular element is the overburden stress. Th is is used to establish the initial values of bulk and shear modulus. The effets of wheel load are then omputed by applying it in 1 equal steps and, finally, iterating until satisfatory onvergene is obtained. A signifiant feature of the proedure is that it is based on the seant modulus at eah step (i.e both stress and strain values are relative to zero). When a onvergent solution has been obtained, the program omputes an equivalent Young's modulus and.poisson' s ratio on the basis of tbe appliation of traffi loading alone. This is essentially a hor<l modulus and is of interest in alibrating simpler linear elasti proedures suh as BlS'rRO 17, pp ) or the Chevron (H. Warren and W. L. Diekman, Numerial Computation of Stresses and Strains in a Multiple-Layer System, unpublished i nternal report, Chevron Researh Corporation, 1963) layered system programs. These deal only with stresses indued by wheel loading and, beause all layers are assumed to be linearly elasti, require an equivalent single value of Young's modulus for the granular layer. The SENOL program was used to determine appropriate values In applying the K-9 model to pavement analysis using the finite element method, the peak stress (overburden plus traffi) is alulated using an assumed initial value. The v alue of e at this peak stress is then omputed and the orresponding restlient modulus is determined from Equation 1. This is regarded as Young's modulus and is ombined with an appropriate value of Poisson's ratio, usually a onstant value, for proeeding with the alulation. The inadequaies of the K-6 model lead to some elements exeeding failure onditions and these are arbitrarily adjusted to bring the stress ondition down to an aeptable level ( 2). l\rbi trary adjustments of this kind are also used in those approahes that adopt a "tension orret ion" for elements in a granular layer. These adjustments are not neessary when using SENOL with the ontour model beause elements approahing failure are automatially assigned low stiffnesses in aordane with the greater detail of this model. Nonetheless, some elements do have final stress onditions just above failure. This is a on-

3 Brown and Pappin 47 sequene of the slight shortomings of the ontour model for stress onditions lose to failure. Computations were performed using the T<- model as well as the ontour model for the two materials noted previously. In applying the K- model, the same basi omputational proedure was followed as for the ontour model: that is, a seant modulus approah with the load applied in stages as desribed previously. In this ase, seant values of E were determined at eah stage of loading and a onstant value of Poisson's ratio (. 3) was adopted throughout. In addition, linear elasti solutions were obtained for several strutures using the BISTRO <ll omputer program. The results from these various omputations were used to study the following points: 1. Stress onditions in the granular layer and the inidene of failure in partiular elements, 2. Comparison of Models A and B for the granular layer, using 5. Equivalent values of Young's modulus, Comparison of ritial parameters SENOL and BISTRO, and Assessment of the K- model. DETAILS OF PAVEMENT STRUCTURES omputed A parameter study was onduted using the SENOL omputer program and it involved omputations on 56 pavement strutures definitions of whih are given in Table 1. Eah struture onsisted of a linear elasti asphalt layer, a nonlinear granular layer, and a linear elasti subgrade. Table l gives the ombinations of stiffnesses and thiknesses that were used. Granular material Model A was adopted for all 56 ases. In addition, six ases, numbered 3 to B in Table 1, were analyzed using Model B and eight ases (1 to B in the table) were also analyzed using the BISTRO linear elasti proedure. Comparisons of the ontour and K-6 models were made using strutures numbered 3 to B. A suary of the eight strutures that were examined in detail is given in Table 2. In eah ase a 4-kN wheel load having a ontat pressure of 5 kpa was used. For the 56 strutures that were analyzed, five solutions did not onverge, some indiated elements at or slightly above failure (q/p' > 2.2), others inluded elements in the zone just below failure (2.2 < q/p' < l.b), and all elements in the remainder were in the region of lower stress levels within whih the ontour model has greatest validity. These various ategories are identified in Table 1, whih shows a trend from the weakest (nononvergent) strutures, through the intermediate areas, to those strong pavements with the lowest peak stress ratios. The signifiane of a nononvergent solution is that a large number of elements within the granular layer are at failure. The general impliation of this is that signifiant permanent deformations are likely to develop in suh a struture. Shaw () has shown that the parameter that determines the tendeny for permanent strain to aumulate under repeated loading is the minimum horizontal distane (value of p') between the end of the stress path and the failure line (see Figure 2). Figure 3 shows, in more detail, the inidene of failure elements in Strutures l and 2 of those investigated in detail. The elements with stress ratios in the transition zone just below failure are also shown. In both these ases the stress onditions in the granular layer are generally high and suh pavements are unlikely to have long lives. Strutures 3 and 4 {Table 2) had some elements in TABLE 1 Modular Ratios Between Granular Layer and Suhgrade h 1 () h2 () E 3 (MPa) Stiffness= 4 GPa , , * o * Stiffness= 7 GPa S.5* * i.s * * Stiffness= 12 GPa * 3. 3 NC 2,5+ so NC 2. 7 NC 1, NC ' 3 5.5* * 2, 7 2.5* 7 2 NC ' so 4.* 7 Note: h 1 =asphalt thikness, h2 = granular thikness, E 3 = subgrade stiffness, =some failure elements, NC = nonovergene-general failure, and +=lements lose to failure; supersripts I to 8 refer to pavement number in Table 2. the transition zone, and the remaining strutures had all elements at stress ratios less than 1.8. The relative potential performane of the granular layer in six ases is refleted by the pavement lives given in Table 2. These were alulated using the pavement evaluation tehniques developed by Brunton and Brown (9) and relate to British onditions. In all ases, exept Pavement 4, the potential failure mehanism was fatigue raking of the asphalt. In Pavement 4 it was exessive rutting. The elasti stiffness of the granular material used in this evaluation, whih is based on linear elasti analysis, was derived from the SENOL omputations. This point is dealt with in the next setion. EQUIVALENT STIFFNESSES FOR THE GRANULAR LAYER Values of Young's modulus and Poisson's ratio are omputed in the SENOL program on the basis of the stresses and resilient strains resulting from traffi loading alone. This "hord modulus" is printed out for eah element together with the orresponding Poisson's ratio. The variation of these parameters through eah of the 56 strutures that were analyzed allows onlusions to be drawn about the equivalent Young's mod-

4 48 Transportation Researh Reord 122 TABLE 2 Details of Pavement Strutures Investigated in Detail Pavement Life Stiffness Thikness No. (msa)" (GPa) () I Granular Thikness Nonlinear Stiffness BISTRO () Model (MPa) Calulation 7 A 3 No 45 A 5 No 7 A,B, K41 2 Yes 2 A,B, K- 7 Yes 7 A,B, K41 3 Yes 2 A,B, K41 5 Yes 5 A,B, K4! 2 Yes 2 A,B, K41 7 Yes 8 msa = millions of standard (8 kn) axles. Load Radial distane () j i 1 Unfalled Zone eoo 7 Transition Zone Stiffness; 3 MP a it.load Radial dlatane () o I 1 oo stiffness; 5 GPa 1 -v-.--,-,, Transition Zone 6 stiffness ;5 MPa Unfalled Zone FIGURE 3 Inidene of failure elements in Pavement& 1 and 2. ulus, or stiffness, that the granular material mobilizes in situ. This information is partiularly useful for allowing seletion of the appropriate single value of granular material stiffness for use with linear elasti layered system programs suh as BISTRO <ll and the Chevron program. Alternatively, it ould form a basis for subdividing the granular layer if this were more appropriate. In the past the approah to defining stiffness for a granular layer has been to use a ertain value of modular ratio between this layer and the subgrade. Values in the range 1.5 to 5 have been generally adopted with 2 the most oon. This approah implies that the stiffness of a partiular granular material adjusts itself in situ in response to the stiffness of the support and the onsequent stress onditions. The SENOL data for all 51 of the strutures, for whih solutions were obtained, were studied and mean values of modular ratio, based on the omputed hord moduli, were extrated. These are given in Table 1 from whih it will be seen that they range from 1.5 to 7.5, a spread similar to that reported from in situ vibration testing (1). However, it will be noted that the high ratios were for the soft subgrade and vie versa, implying that the atual stiffness of the granular layer does not vary greatly. To produe reliable values of these dedued equivalent stiffnesses for the granular layer, only those strutures with peak stress ratios below 1. 8 (well below failure) were onsidered. This redued the number of relevant solutions to 22 as an be seen from Table 1. Beause a value of hord modulus is omputed for eah element, variations within the struture were studied. Within a radius of 35 from the load enterline the variation of this parameter and the assoiated Poisson's ratio were quite small, Figure 4 illustrates this point for Pavements 5 and 6 of those analyzed in detail (Table 2). The shaded zones over 'the range of values up to a radius of 35 and results are shown for both granular material Models A and B. There is a general trend for stiffness to inrease slightly with depth in eah ase. However, this variation is suffiiently modest to onsider a single equivalent value of stiffness for the layer as a whole, when ontemplating linear elasti layered system alulations. For Model A, the mean equivalent stiffness for the 22 strutures under onsideration varied from 6 to 125 MP a. These values are small in relation to the stiffnesses of the asphalt layers (4 to 12 GPa). It was, therefore, onsidered appropriate to use a single value of 1 MPa in pavement design alulations based on linear elastiity and involving good quality granular subbases. This was adopted for the Nottingham analytial design proedure (9), For Model B, representing poorer quality material, only six pavements were studied (3 to 8 in Table 2) with stress levels well below failure. The range of mean stiffnesses was 35 to 5 MPa and a mean value of 4 MPa is suggested for routine design. For both models, Poisson's ratio was.3 to.4, the former value having been adopted for design.

5 Brown and Pappin 49 1 Stillness (MPaJ Stillness. 4 GP a Model A 1 Stiffness (MPa) -+-5Lo1 1_5._o Stillness= 4 GPa 'E E - u.! u :i en, - "C 1..8 al iii.6 al.4 :::;.2 Line of equality 4 5 Sub grade Stiffness= 5 MPa Surfae defletion () Model B Stillness 3 MP a FIGURE 4 Variation of hord modulus (stiffness) through the granular layer. '2 " u "! Oi.. ( :: 2 :2 a; al iii 18 ;;; 16 ::::i EVALUATION OF LINEAR ELASTIC SOLUTION Against the foregoing bakground, a number of alulations were arried out using the BISTRO omputer program so that omparisons ould be made with SENOL results for ertain ritial parameters. The parameters seleted were tensile strain at the bottom of the asphalt layer (the fatigue raking design riterion) and surfae defletions at radii up to 35 as an indiation of overall pavement response. Model A material was used and a mean stiffness of 1 MPa was adopted for the granular material in the BISTRO omputations. Figure 5 shows omparisons that generally indiate that the linear elasti layered system approah produes quite reasonable values for these two parameters, although surfae defletions omputed using BISTRO are somewhat high. There was no signifiant differene in the defletion omparisons at different radial positions. In reality, not only is the granular layer nonlinear, so is the subgrade. A few alulations were onduted with a nonlinear elasti model for the subgrade (5) derived from work by Brown et al. (11). The results, based on a linear elasti subgrade, were ompatible with those disussed in this paper. STRESS CONDITIONS IN GRANULAR LAYERS The foregoing setion has shown that linear elasti layered system omputations an determine ritial design parameters when an appropriate equivalent stiffness is assigned to the granular layer. They are unlikely, however, to be able to reliably alulate stress onditions within the granular layer itself. One of the partiular problems in this onnetion is the tendeny for tensile stresses to be apparent in granular layers when linear elasti assumptions are used. This point is illustrated in Figure 6 for tensile strain (mirostrain) FIGURE 5 Comparison of results from SENOL and BISTRO omputations. Pavements 3 and 4 (see Table 2), and the results from BISTRO alulations are ompared with those from SENOL using Model A material. The left line of eah pair represents the traffi-indued horizontal stress in the granular layer, and the right line shows the influene of the ompressive overburden pressure. For linear elastiity, even when overburden is inluded, tensile stresses still result in the lower half of the layer. Similar analysis for Pavements 5 and 6, whih were stronger, showed that these ombined stresses an beome ompressive in favorable irumstanes. However, by ontrast, the SENOL results show ompressive stresses in all these ases and in most others as well. The inidene of tensile stress in a granular layer does generally imply a failure ondition. Ho,1- ever, failure is defined by the stress ratio (q/p'), whih is influened by vertial stresses as well as horizontal ones. Tensile total stresses, in soil mehanis terms, may orrespond to ompressive effetive stresses if the granular material is subjet to negative pore pressure, whih in general it will be, However, quantifiation of this pore pressure may not be easy. Table 3 gives the peak q/p' ratios determined at the top and bottom of the granular layer for Pavements 3 to 6. The SENOL values range from O. 8 to 2.1, all below the failure ondition of 2.2, and in only two ases are the BISTRO values below failure. These data, therefore, onfirm the point that detailed study of stress onditions within granular layers annot be undertaken using linear elasti theory.

6 5 Transportation Researh Reord 122 Horizontal stress (kpa) o.---'-- -L--_., ---'---'----'-- A s p ha 11 Stillness =7 GPa Dphr r---=;::::=7" Sub grade Pavement 3 Horizontal stress (kpa) Stillness= 2 MPa '----'' '--'--'- Stiffness= 7 GPe Dh r--t--,::--:-- 2 n-llneer 4 Pavement 4 Stillness= 7 MPs FIGURE 6 Horizontal stresses in granular layer. TABLE 3 Stress Ratios at Top and Bottom of Granular Layer q/p SENOL/Contour BISTRO Pavement No. Top Bottom Top Bottom EVALUATION OF K-8 MODEL The K-8 equation for the well-graded rushed limestone was used in the SENOL program with a seant modulus approah for Pavements 3 to 8 (Tahle 2). The maximum tensile strain in the asphalt and surfae defletions up to a radius of 35 were extrated from the output for omparison with the ontour model results. Figure 7 shows that the defletions ompare favorably. However, the K-8 approah underpredits the tensile strain and is less satisfatory than the linear elasti layered system solutions (Figure 5), whih used a single value of stiffness for the granular layer. Al though the K-8 model may be of use in evaluating effets in other layers, the results showed that stress onditions in the granular layer are not orretly determined. This point is illustrated by Figure 8, whih shows substantial numbers of failure elements in Pavements 5 and 6 that were analyzed using the K-8 approah, whereas no failure elements were predited using the ontour model. CONCLUSIONS 1. A detailed study of the strutural behavior of unbound granular materials in pavements requires an aurate stress-strain model to define nonlinear elasti response i.8!, 'i.6.! :f ii, CD I :.I:.4 ;; '.2, 'iii Surfae defletion (m) :--''\ '11" 18 e o' <1', 16 - :f CD o. I :.I: 14 -Oi 12.. "' tensile strain (mirostrain) FIGURE 7 Comparison of results from ontour and K-6 models. 6 8 Load rt_ I Radius () Stillness= 3 MP a Pavement 5 Load Ci. I Radius () Stiffness= 3 MP a Pavement 6 FIGURE 8 Inidene of failure elements in Pavements 5 and 6 using the K-6 model. 2. The ontour model published by Brown and Pappin and the assoiated SENOL finite element omputer program allow this to be done, but improved modeling is still desirable. 3. The SENOL program and the ontour model allow equivalent elasti stiffnesses for granular layers to be determined for use in layered system analysis. 4. The onept of a fixed modular ratio between

7 Brown and Pappin 51 a granular layer and the subgrade appears inappropriate beause a single value of stiffness, dependent on the granular material, may be used in linear elasti analysis to determine effets in other layers of the struture for pavement design and evaluation purposes. 5. A well-graded rushed 1 imestone base has an equivalent stiffness of 1 MPa, whereas a poorly graded material has a stiffness of only 4 MPa for the range of onditions investigated. 6. Linear elasti layered system programs an determine surfae defletions and maximum asphalt tensile strains to an aeptable auray for design when the orret equivalent stiffness is assigned to the granular layer. 7. The finite element method inorporating the K-9 model an be used to determine surfae defletions and asphalt tensile strains but is unable to determine the stress onditions within the granular layer. 8. Conlusions 6 and 7 suggest that the simplest approah to design alulations for surfae deflet ion or asphalt tensile strain involves the use of linear elasti layered systems, provided the orret equivalent stiffness is defined from detailed nonlinear finite element analysis. 9. Design omputations involving deformation or failure within the granular layer require a detailed model and finite element analysis. ACKNOWLEDGMENTS This study formed part of a projet sponsored by the European Researh Offie of the U.S. Army and ICI Fibres. Related work under ontrat to the Transport and Road Researh Laboratory and British Rail made the omputations possible. The authors are grateful for this support and the assistane of their olleagues, P. Shaw and Janet Brunton, who performed some of the alulations. The failities of the nepartment of Civil Engineering under R.C. Coates and, subsequently, P.S. Pell, were made readily available, and the servie of the Cripps Computing Centre at the University of Nottingham is also aknowledged. 2. L. Raad and J.L. Figueroa. Load Response of Transportation Support Systems. Journal of the Transportation Engineering Division, ASCE, Vol. 16, No. TEl, 198, pp J.L. Figueroa and M.R. Thompson. Simplified Strutural Analysis of Flexible Pavements for Seondary Roads Based on ILLI-PAVE. In Transportation Researh Reord 766, TRB, National Researh Counil, Washington, D.C 198, pp S.F. Brown and J.W. Pappin. Analysis of Pavements with Granular Bases. In Transportation Researh Reord 81, TRB, National Researh Counil, Washington, D. 1981, pp S. F. Brown and J. W. Papp in. Use of Pavement Test Faility for the Validation of Analytial Design Methods. Pro Fifth International Conferene on the Strutural Design of Pavements, Delft, The Netherlands, Vol. 1, 1982, pp J.w. Pappin and S.F. Brown. Resilient Stress Strain Behaviour of a Crushed Rok. Pro International Symposium on Soils under Cyli and Transient Loading, Swansea, England, Vol. 1, 198, pp M.G.F. Peutz, H.M.P. Van Kempen, and A. Jones. Layered Systems under Normal Surfae Loads. Highway Researh Reord 228, HRB, National Researh Counil, Washington, D.C 1968, pp P. Shaw. Stress-Strain Relationships for Granular Materials under Repeated Loading. Ph.D. dissertation. University of Nottingham, England, J.M. Brunton and S.F. Brown. Computer Programs for the Analytial Design of Pavements. Highways and Transportation, Vol. 31, No. 8/9, 1984, pp W. Heukelom and A.J.G. Klomp. Dynami Testing as a Means of Controlling Pavements During and After Constrution. Pro International Conferene on the Strutural Design of Pavements, Ann Arbor, Mih 1962, pp S.F. Brown, A.K.F. Lashine, and A.F.L. Hyde. Repeated Load Triaxial Testing of Silty Clay. Geotehnique, Vol. 25, No. 1, 1975, pp REFERENCES 1. C.L. Monismith, H.B. Seed, F.G. Mitry, and C.K. Chan. Predition of Pavement Defletions from Laboratory Repeated Load Tests. Pro Seond International Conferene on the Strutural Design of Pavements, 1967, pp Publiation of this paper sponsored by Coittee on Mehanis of Earth Masses and Layered Systems.