Backcalculation of Design Parameters for Rigid Pavements

Transcription

1 TRANSPORTATON RESEARCH RECORD Bakalulation of Design Parameters for Rigid Pavemts J. UZAN, R. BRGGS, AND T. SCULLON Currt rigid pavemt bakalulation proedures use only interior or ter slab defletion bowls that are measured far from any rak or joint. Usually two parameters-the rigidity of the slab and the modulus of subgrade reation-are evaluated. The bakalulated subgrade modulus is reommded for design subjet to a orretion/alibration fator. However, sine the modulus of subgrade reation depds on the size of the loaded area, various alibration fators may be required for various loading onditions at the time of the testing and for various assumptions in the design proedure onerning the loading ondition, namely interior or edge loading onditions. Furthermore, wh a stabilized layer is used beath the onrete slab it is not lear what the bakalulated parameters represt. Does the.bakalulated rigidity of the slab inlude the stabilized layer or is it inluded in the bakalulated subgrade modulus of reation? The answers to these questions have an impat on the omputation of stresses that are the basis of any rigid pavemt design proedure. A proedure for bakalulating the slab rigidity and the modulus of subgrade reation for either the ter slab or the free-edge loading ondition is briefly desribed. The material parameters and the slab rigidity were bakalulated using the plate on Winkler foundation for interior and free-edge loading onditions. The ter slab results are ompared with those that are bakalulated using linear elasti layer models. The proposed method uses the MODULUS omputer program framework in whih a data base of theoretial defletion bowls is gerated and used thereafter with a pattern searh algorithm to find the set of parameters (the harateristi lgth e and the subgrade modulus of reation k) that minimizes the error betwe measured and omputed defletions. Results bakalulated from five Strategi Highway Researh Program jointed onrete monitor sites are prested. The slab thiknesses range betwe 9.3 and 11.4 in., and the base layers are either granular, asphalt, or emt stabilized. Defletion tests were made at three load levels using a falling weight defletometer. Results inlude (a) a omparison of C and k values bakalulated at the ter slab and at the edge (it is learly se that the k-values derived from free-edge ondition are two to four times larger than those obtained from ter slab defletions) and (b) a omparison of results omputed using both the Hogg model and the linear elasti layered model. t is learly se that the Hogg model produes subgrade moduli similar to those produed using the linear elasti model. Furthermore, with the Hogg model the effet of the stabilized base layer is inluded in the C-parameter. more omplex than those in flexible pavemts. These onditions are best treated with finite elemt solutions of plates resting on one- or two-layer systems or on a Winkler foundation. Wh load transfer harateristis are inluded in the evaluation, the problem beomes too ompliated for standard bakalulation proedures. The approah is prested in Uzan (1) and some is repeated here for ompletess. The bakalulation sheme makes use of simple solutions that assume that the raks or joints are far from the load and do not affet the defletion bowl. These solutions are appliable for plain or simply reinfored onrete pavemts in whih raks are widely separated, in ontrast to ontinuously reinfored onrete pavemts in whih raks are fairly lose. n geral, the distane betwe joints and raks would be less than eight times the harateristi lgth of the system (in whih ase the size of the plate may be onsidered as infinite (2,3)], and the use of the simple solutions onstitutes an approximation. The framework prested in the paper is geral and finite elemt programs instead of the simple solutions ould be used for defletion omputations. n this paper a bakalulation proedure is desribed to determine the appropriate layer properties from field defletion measuremts. The proedure involves two steps: in the first step, a fatorial defletion data base is built using multiple runs of a theoretial model; in the seond step, a pattern searh and interpolation sheme are used to math field measured and theoretial defletion bowls. The output is the set of layer properties that minimizes the error betwe measured and omputed defletions. n this paper the authors ompare the results from three differt theoretial pavemt models: (a) a multilayer linear elasti model; (b) Hogg model, and Hertz-Westergaard model. Furthermore, alulations with defletions measured near a free edge an be ompared with those made with defletions measured at the ter of the slab. This analysis ables an examination of the signifiane of the k-subgrade modulus of reation and the proedures of modeling rigid pavemts with stabilized base layers. The evaluation of the load-deformation harateristis from defletion-based nondestrutive defletion testing (NDT) is, in esse, similar for flexible and rigid pavemts (1). However, beause of the existe of joints, raks, and free edges, the boundary onditions prevailing in rigid pavemts are J. Uzan and T. Sullion, Texas Transportation nstitute, Texas A&M University, College Station, Tex R. C. Briggs, Texas Departmt of Transportation, 125 East 11th Street, Austin, Tex THEORETCAL MODELS FOR COMPUTNG SLAB DEFLECTONS Three types of models are used for evaluating the loaddeformation harateristis of pavemt materials. These are the multilayer linear elasti model, the Hogg model of a slab supported by an elasti foundation, and the Hertz-Westergaard model of a slab supported by a liquid foundation. These models all use linear load-deformation harateristis, implying on-

2 18 stant moduli of elastiity, Poisson's ratios, and modulus of subgrade reation. n the multilayer elasti system model, the pavemt is represted by linear elasti layers that extd to infinity in the horizontal diretion. This model, widely used in flexible pavemt evaluation, orresponds to the interior loading ase in whih both the load and the defletion ssors are far from any joint or rak. The boundary onditions at the interfaes from the layers an be varied from ompletely rough to smooth (4,5). n the ase of a onrete pavemt with unbound base material, the interfae onditions have negligible effets on deformations and stresses. However, wh the base layer is stabilized, the effets of interfae onditions on stresses and deformations may be signifiant. The seond model onsidered is the Hogg model (6,7). With this model the pavemt is represted by an infinite slab resting on an elasti foundation. This model, like the linear elasti model, orresponds to the interior loading ase only. The model is in esse a two-layer system with additional assumptions onerning the first layer, whih is represted by a slab (8). n this model, the pavemt parameters are the subgrade modulus of elastiity, Esv and the radius of relative stiffness of the slab subgrade system ee (whih is a funtion of the elasti parameters of the onrete and of the sub grade and the slab thikness). The third model used is.the Hertz-Westergaard model (9,1). With this model the pavemt is represted by an infinite or semiinfinite slab resting on a dse liquid foundation (also known as a Winkler foundation). n this model, two loading ases are onsidered: (a) the interior loading ase (known as the Hertz model) in whih the slab is infinite and the load and defletion ssors are far from any joint or rak; and (b) the free-edge ase (known as the Westergaard model) in whih the slab is semiinfinite and the load and defletion ssors are near a free joint and far from any additional joint or rak. The pavemt parameters are the subgrade modulus of reation, k, and the radius of relative stiffness of the slabsubgrade system ek (whih is a funtion of the elasti parameters of the onrete, the sub grade modulus of reation and the slab thikness). Closed-form solutions for defletion bowls exist only for the ase of a ontrated load (9,11). n the ase of a uniform pressure distributed over a irular area, the solution of the ontrated load may be integrated numerially. n the interior loading ase, a omputer program giv by Selvadurai (11) was used to ompute. the defletions at the surfae. n the free-edge loading ase, Westergaard equations (12) were integrated over the loading area using the proedure prested by Uzan and Sides (13). n the three-layer and Hogg models, a rigid base at any depth in the subgrade an be easily inluded. The Hertz-Westergaard model does not have this apability beause the subgrade is represted by a spring or a liquid that is dimsionless. BACKCALCULA TON PROCEDURE The bakalulation of layer properties for rigid pavemt follows the MODULUS bakalulation framework already developed for flexible pavemts. t is desribed in detail elsewhere (14,15). For flexible pavemts the outputs are the TRANSPORTATON RESEARCH RECORD 1377 set of layer moduli that minimize the error betwe measured and theoretially alulated defletion bowls. The existing system is a two-step proess. n the first step, a linear elasti program is run for a range of layer moduli, and the resulting defletions are stored in a defletion data base. n the seond step, a pattern searh and interpolation sheme are used to minimize the error betwe measured and omputed defletion bowls. The multilayer system model desribed in this paper is esstially the same as the flexible pavemt system. The dat base is built by varying the moduli of eah pavemt layer in tum and storing the omputed defletions. n both the Hogg and the Hertz-Westergaard models, the data base is built by varying the subgrade modulus 58 (or the subgrade modulus of reation k) and the radius of relative stiffness ee (or fk). n the analysis disussed for the Hogg model, 1 values of subgrade modulus (range 2, to 75, lb/in. 2 ) and 1 values of ee (range, 1 to 21 in.) were omputed. This resulted in 1 theoretial defletion bowls being gerated and stored. For Hertz-Westergaard, a 1 x 1 fatorial was again stored with ranges of k for 25 to 1,5 lb/ in. 3 and ek from 1 to 14 in. These ranges are intded to represt the limits of possible values. The program an handle any ssor arrangemt. n all ases defletions are omputed at the ssor loations used in the field measuremts. For the SHRP testing program this is at, 8, 12, 18, 24, 36, and 6 in. from the ter of the load. Only one data base of omputed surfae defletions is required for eah ssor arrangemt, and the first step of the proedure need not be repeated if the data base is already available. The proedure used in this paper an handle any number of ssors or any ssor arrangemt. Moreover, the Hogg model inludes both the infinite and finite subgrade thikness ases. The proedure prested is unique in that both ter and free-edge defletion bowls an be analyzed to bakalulate the modulus of subgrade reation and the modulus of elastiity of the onrete. The results of the numerial integration have be heked against other analysis methods, suh as the FE results prested in the appdix, and have be found to be omparable. A diret omparison of the results of bakalulation obtained with the proedure desribed in the paper (named JUSLAB) and with the LL-BACK omputer program follows. Table 1 prests results of bakalulation for eight rigid pavemt setions at three differt load levels. The proedures and objetive funtions of the bakalulation are usually differt in JUSLAB and LL-BACK programs. For the sake of omparison, the proedure and objetive funtion in JUSLAB were hanged to resemble those in LL-BACK. n LL-BACK, the omputation proedure is (a) to obtain the harateristi lgth from the basin area; (b) to ompute a k-value for eah ssor; and to ompute a mean value. t appeared that in several ases analyzed the omputed k-value from the last ssor defletion was muh lower than the other k-values obtained from the other six ssors. The proedure of taking the mean is th similar to dropping the last ssor. The results of bakalulation using JUSLAB were obtained by fitting the first six ssors (by means of weighting fators) and minimizing the squared error betwe measured and omputed defletions at these six ssors. The results obtained with JUSLAB and LL-BACK ompare very well; differt

3 Uzan et al. 19 TABLE 1 Comparison of Bakalulation Results Subgrade Modulus of Reation in pi, for Load Level Setion JU SLAB LL-BACK JU SLAB LLl-BACK JU SLAB LLl-BACK J K L M N p results will be obtained if the sevth ssor is not dropped from the analysis and if the objetive funtion to be minimized is hanged, for example, from squared error to absolute error. CASE STUDY DESCRPTON To evaluate the bakalulation proedure, Strategi Highway Researh Program (SHRP) falling weight defletometer (FWD) data were proessed on five SHRP geral pavemt study sites in Texas. Average slab thiknesses and other information are giv in Table 2. All the pavemts were jointed onrete with asphalt shoulders. n the SHRP defletion testing proedure, defletions are tak at midslab, edge, omer, and pavemt joints. n this analysis the ter slab and the edge FWD defletion data were used. SHRP uses three loading levels for rigid pavemts (approximately 9, 12, and 16 kips) and ollets four repliate drops at eah test point. n the analysis that follows the four repliate drops were normalized to the average load and th averaged so that a single defletion bowl was proessed for eah load level. This proedure takes are of the random errors, as planned by SHRP. Therefore, the averaging proess used inludes (a) omputing the average load, (b) normalizing eah of the four defletion bowls to the average load, and averaging the normalized bowls. n geral, 6 defletion bowls were proessed for eah site, three load levels at 2 positions. n all ases the load was applied on a 12-in.-diameter FWD load plate. Defletion ssors were loated at, 8, 12, 18, 24, 36, and 6 in. from the ter of the plate. PRESENTATON OF RESULTS The results of this analysis are shown in Figures 1 through 5, details of whih follow. The results from all five sites are prested on a single diagram. Within eah site the results from various loations and load levels are prested. n eah analysis all sev ssors were used in mathing measured TABLE 2 Layer Charateristis of Study Pavemts SHRP D # Pavemt Charateristis Plain Reinfored Reinfored Reinfored Reinfored Joint Spaing 15 ft 15 ft 6-2 ft+ 6-2 ft+ 3 ft++ Slab Thikness (nhes) l 1.l Base Subbase Type* Thikness (nhes} Type* Thikness (nhes} Subgrade AC 3.7 LT 7.2 Clay GR Sandy/ Clay CT 5.8 LT 6. Sandy/ Clay AC 7.9 SS 8.8 Sand CT 6.3 LT 5.6 Sandy/ Clay AC Asphalti Conrete GR Crushed Gravel CT Cemt Treated LT Lime Treated SS Selet Sand + 6 ft 6 in betwe onstrution joints with towels 2 ft betwe warping joints. ++with transverse raks in eah slab.

4 11.and theoretial bowls, and the average error per ssor was omputed. Figure 1 shows the pert error per ssor betwe measured and omputed defletion. t is se that the error is very large, approahing 1 pert in the ases of the Hertz and Westergaard models ompared with the Hogg or layered linear elasti model. n the following analysis, all solutions that resulted in more than 6 pert error per ssor (and solutions that resulted in unaeptable results) have be exluded for Figures 2 through 5. Figure 2 shows the results for the linear elasti layered analysis in terms of the bakalulated modulus of the subgrade and modulus of onrete at ter slab. n modeling the pa emt, the omplete layering shown in Table 2 was used. For example, for setion 4152 the pavemt was modeled as four layers (11.4 in. of onrete resting on a 6.3-in. emt-treated base over a 5.6-in. lime-treated subbase over a semi-infinite depth subgrade). n all modeling, the subgrade depth was set at semi-infinite. The bakalulated subgrade moduli ranged from 2 to 45 ksi and onrete moduli ranged from 4 to 8 million lb/in. 2 (8 million lb/in. 2 was set as maximum). Figure 3 shows the bakalulated results from using the Hogg model on the ter slab defletions. The subgrade was TRANSPORTATON RESEARCH RECORD 1377 assumed to be semi-infinite. The bakalulated subgrade moduli ranged from 2 to 5 ksi and harateristi lgth ee ranged from 25 to 45 in. Figure 4 shows the bakalulated results obtained using the Hertz model with the ter slab defletions. The modulus of subgrade reations ranged typially from 5 to 2 lb/in. 3, with a few readings reahing 3 lb/in. 3 The harateristi lgths ranged typially from 4 to 7 in. Figure 5 shows the bakalulated results obtained using the Westergaard edge solution. The modulus of subgrade reations ranged typially from 2 to 7 lb/in. 3 with harateristi lgths ranging from 35 to 6 in. DSCUSSON OF RESULTS The following observations are made with regard to the results prested in Figures 1 through Ev after sreing, the error in the fitting of the defletion bowl seems to be very large in the Hertz and Westergaard models. However, the error is aeptable for the 1 8- FWD Load&..- Low.p + MMUm t:z kip ti.p )( g CD Cl e 2 e CL. 8. g Cl) CZ> C) e , (a) (b) (d) FGURE 1 Average error per ssor betwe omputed and measured defletion: (a) linear layered elasti model; (b) Hogg with ter defletions; Hertz with ter defletions; and (ti) Westergaard with edge defletions.

5 Uzan et al u;. 4 o; "O! Ui' C> "O JJ as :l llj Q 3 :l (/) E. :l '3 "O 2 E 1 33 i., i ll..!! FWD Load Level Low 9 kip + Medium Z kip.. ltgll 111 kip 4152 u;.. 8 )1( )( 6! u u (/) :l 4 '3 "O E ll! )(P i+ i.,).( +: +>< )( f'<+ +1 i +., i' 1! )( 1.. +!lot+ P.! >t r-><)(. x ll )( $ + +. oti xi ><! )( )( FGURE 2 Subgrade (top) and onrete (bottom) modulus bakalulated using linear layered elasti model for eah SHRP test setion. t )( FGURE 3 Subgrade modulus (top) and slab harateristi lgth (bottom) bakalulated using Hogg model with ter slab defletions. Hogg and layered linear elasti models. n the ase of rigid pavemts, material behavior is very lose to linear and does not indue appreiable systemati error. t may be argued that the error is aused by the boundary onditions that are not fulfilled beause of the proximity of the raks and joints. However, this defiiy exists for both the Hertz and Westergaard models, as well as for the Hogg and layered linear elasti models. t is the authors' opinion that the bad fitting is the result of systemati errors indued by the model represtation of the subgrade support, namely, that in the ases of strong subgrade, with elasti moduli of 2 to 45 ksi, the subgrades annot be simulated as liquid or spring. The above hypothesis should be heked with weaker subgrade materials. f this is found to be the ase, the use of the k model -should be questioned in rigid pavemt design. The bakalulation was made using all ssors, inluding the sevth ssor, whih was found in several ases to be underpredited. A similar trd is found in LL-BACK results in whih the error aused by the sevth ssor may be in exess of 3 pert. 2. The modulus of subgrade reation at the edge is two to four times higher than that obtained from the ter slab. A disussion of this finding together with a omparison of earlier results, suh as those of Teller and Sutherland (16),- is giv in the appdix to the paper. The k-values at the ter of the slab appear very low. As will be disussed later, the modulus of subgrade reation represts the subgrade layer and does not inlude the stabilized base material on top of the subgrade. This result suggests that the modulus of subgrade reation is not an adequate material property for modeling slab response beause it is a funtion of the loading geometry. Using linear elasti theory, one an find that the subgrade modulus of reation is proportional to the modulus of elastiity and inversely proportional to the size of the loaded area. 3. From Figures 2 and 3 it is observed that the modulus of the subgrade is idtial for both the linear layered elasti model and the Hogg model. The Hogg model is a simplified two-layered system, whereas the linear elasti is either a threeor four-layered system. This differe implies that in the Hogg model, the base-subbase layers are inluded in the plate rigidity to give a omposite plate that rests on the subgrade. This result is very important for hoosing the omputation sheme of the bding stresses. n this ase the method of equivalt momt of inertia or thikness should be used in

6 (3 FWD Load Level. 3 Low ldpa + Medium 12 kip.. ltgll 11 kip! 25..;.,. J k '..-.i "' T 2 sdllf. H ::::s, " =- ::::s "5 1 i "8 E ii lw A*! TRANSPORTATON RESEARCH RECORD r---..-r---, 8. "'"'""'"'" :.::: kip -----t-----;ip----;.2.. ltgll 11 kip j 71.,.g '*'1-:=----d. >< -... o S!liil._:.,,_,,,. till<.. :.1...z-_ lla---l-lllt't.l!!ik---1 " "& E j i l 9 8.s::. :i< + " +.5.s 7 iillli Cl J i "" )( )( )(., + )(.! mx )( i 6 >t "C + i 5 as.s::. 4 3 FGURE 4 Modulus of subgrade reation (top) and slab harateristi lgth (bottom) bakalulated using Hertz model with ter slab defletions. omputing the bding stresses. Similar behavioral effet is assumed to our for the Hertz-Westergaard model, namely that the stiffness effet of the base and subbase layers is inluded in the plate (pavemt) rigidity (1). 4. n all ases the e values bakalulated <1.re relatively large. However, as mtioned earlier, it appears that this value inludes a ontribution from the base and subbase layers. To evaluate this ontribution the equivalt thikness he 8 for the slab was alulated using the following formula: h 3 = et: 12(1 - vdk eg Ee (1) for the Hertz-Westergaard model and h3 = 2C 12(1 - vd e; eg Ee (2) C = E2 (1 - Vz) (1 + v 2 ) (3-4v 2 ) for the Hogg model, (3) ao s::. x>s '6> ';f x.! )( :;::: +.. )( )( : s 4 f!... )C as i )(.s::. + 2 " + )(, 4 i FGURES Modulus of subgrade reation (top) and slab harateristi lgth (bottom) bakalulated using Hertz Westergaard model with edge defletions. where Ee = modulus of elastiity of onrete (an average value of 6,5, lb/in. 2 was used in the alulation), E 2 = modulus of elastiity of subgrade, v = Poisson's ratio of onrete (a value of.2 was used), and v 2 = Poisson's ratio of subgrade (a-value of.4 was used). The equivalt thikness was omputed for eah loation and load level. n Table 3 the average equivalt thiknesses for eah setion are ompared with the atual thiknesses. The equivalt thikness is muh larger than the atual onrete thikness (exept for the seond setions in whih a lower Ee should have be used). t approahes the ombined thikness of the onrete and the emt-treated base together. This result indiates that there is a good bond betwe the layers. To jerive a relationship betwe equivalt thikness heq and pavemt variables, it is important to know the exat layer thikness and to ollet data on a wider range of pavemt setions. 5. The relationship betwe subgrade elasti modulus Es 8 and the subgrade k-value was developed by the Corps of

7 Uzan et al. 113 TABLE 3 Comparison of Atual Slab Thikness with Calulated Equivalt Slab Thikness Atual Slab Setion Thikness Type D (inhes) AC GR CT AC CT Base Thikness (inhes) Equivalt Slab Thikness Figure 3 Figure 4 Figure 5 Hogg Hertz Westergaard (ter) {ter) (edge) t ample) that uses the same infinite slab size onditions. This result makes the edge ondition bakalulation neessary for use with design proedures that uses the edge onditions. 3. Bding stress omputations should be made using the method of equivalt thiknesses. t is reommded that the defletion bowls and the bakalulation method be used with SHRP data to develop and alibrate the method. ACKNOWLEDGMENTS i modulus of elastiity, ksi FGURE 6 Relationship betwe modulus of elastiity and modulus of subgrade reation. Engineers (17) and is shown in Equation 4 (kin pounds per ubi inh and 58 in pounds per square inh): l k _ (log ) og (4) This equation is shown graphially in Figure 6 together with the orrelations obtained from edge and ter slab onditions. As an be se, the Corps of Engineers line lies betwe the lines gerated for edge and ter slab and therefore represts an average orrelation. CONCLUSONS AND RECOMMEND A TO NS The use of four differt models to bakalulate rigid pavemt material parameters has led to the following onlusions: 1. The Winkler foundation model for rigid pavemts may be not appropriate for the ases analyzed with strong subgrades. Similar analyses should be onduted on weak subgrades to further evaluate this onlusion. 2. The subgrade modulus of reation does not seem to be a material property. The k-value for edge onditions is two to four times larger than that for ter onditions for the same site. One should th use the values obtained only for the same onditions that prevailed in their derivations. For example, the values obtained for the ter onditions an be used only with a proedure (design proedure, for ex- This work is sponsored by the Texas Departmt of Transportation. Jerome F. Daleid of Brt Rauhut Engineering supplied the defletion data and pavemt layer data used in the study. APPENDX The depde of the bakalulated k on the loation of load (ter versus free edge) may be questioned, aording to earlier results. For example, on the basis of four slabs and one set of basin measuremts on eah slab (using an average of 12 observations), Teller and Sutherland (16) onluded, "for onditions that are omparable there is rather good agreemt betwe the values of modulus of subgrade reations, k, as determined by pavemt defletion, for the interior and edge loadings but the value for the orner loadings is onsisttly lower." However, Carlton and Behrmann (18) stated, "from the data obtained in this study, it appears that the effetive subgrade modulus is the same for both orner and edge loadings. n previous model tests, however, determinations of the subgrade modulus for the model had shown that for interior loadings, the effetive k was 35 pounds per ubi inh. This is approximately half the value of k measured at the edge and at the orner. t is believed that this appart inrease in k near the boundaries of the slab may be explained by additional support derived from the subgrade outside the limits of the slab." Despite the above ontraditory results, it appears that a gradual inrease in the subgrade modulus as the load moves from the ter to the edge and th to the orner would be aeptable [see work by oannides et al. (3)]. The question would th be, How muh inrease would be aeptable? The results prested in the paper suggested that the ratio of the

8 114 bakalulated k values from edge and interior loading an be as large as 2 to 4. As mtioned in the paper, these results are based on field defletions olleted by the Long Term Pavemt Performane program of SHRP. The differe betwe Teller and Sutherland and SHRP defletion results is se very learly from the ratios of the maximum defletions per unit load in the ter and edge loadings. n the ase of Teller and Sutherland, the ratio is about 1 to 4, whereas in the SHRP setion ase, the ratio is about 1 to 1.3. n the ase study reported by Uzan (1), the ratio of the edge to ter loading defletions did not exeed 2. t is not lear what the ause is of the disrepany of the defletion ratios. t may be attributed to any of the following: the dynami FWD loading as ompared with the stati loading ondition used by Teller and Sutherland, warping of the plates, or the prese of the voids. Readers are giv the opportunity to make their own judgmt about whih value of k to use in the design. REFERENCES 1. J. Uzan. Rigid Pavemt Evaluation Using NDT-A Case Study. ASCE, in press (in preparation). 2. A. M. loannides, M. R. Thompson, and E. J. Barberg. Finite Elemt Analysis o.f Slabs-on-Grade Using a Variety of Support Models. Pro., 3rd nternational Confere on Conrete Pavemt Design and Rehabilitation, Purdue University, West Lafayette, nd., 1985, pp A. M. oannides, M. R. Thompson, and E. J. Barberg. Westergaard Solutions Reonsidered. n Transportation Researh Reord 143, TRB, National Researh Counil, Washington, D.C., 1985, pp D. L. De Jong, M. G. F. Ptz, and A. R. Korswag. Com puter Program BSAR: Layered Systems Under Normal and Tangtial Surfae Loads. External Report AMSR Koninklijke/Shell Laboratorium, Amsterdam, the Netherlands, J. Uzan. The nflue of the nterfae Condition on Stress Distribution in a Layered System. n Transportation Researh Reord 616, TRB, National Researh Counil, Washington, D.C., 1976, pp TRANSPORTATON RESEARCH RECORD A. H. A. Hogg. Equilibrium of a Thin Plate, Symmetrially Loaded, Resting on an Elasti Foundation of nfinite Depth. Philosophial Magazine, Vol. 25, No. 168, 1958, pp A. H. A. Hogg. Equilibrium of a Thin Slab on an Elasti Foundation of Finite Depth. Philosophial Magazine, Vol. 35, No. 243, 1944, pp A. M. oannides. Dimsional Analysis in NDT Rigid Pavemt Evaluation. Journal of Transportation Engineering, ASCE, Vol. 116, No. 1, Jan., 199, pp H. Hertz. On the Equilibrium of Floating Elasti Plates. n Misellaneous Papers by Heinrih Hertz (authorized English translation; (D. E. Jones and G. A. Shott, translators). MaMillan Co., Ltd., London, England, 1896, pp H. M. Westergaard. New Formula for Stresses in Conrete Pavemts of Airfields. Transations, ASCE, Vol. 113, 1948, pp A. P. S. Selvadurai. Elasti Analysis of Soil-Foundation nteration, Developmts in Geotehnial Engineering, Vol. 17. Elvesier, Amsterdam, the Netherlands, G. Pikett and G. K. Ray. nflue Charts for Conrete Pavemts. Transations, ASCE, Vol. 116, 1951, pp J. Uzan and A. Sides. The Effet of Contat Area Shape and Pressure Distribution on Multilayer System Response. n Transportation Researh Reord ll17, TRB, National Researh Counil, Washington, D.C., 1987, pp J. Uzan, R. L. Lytton, and F. P. German. Geral Proedure for Bakalulating Layer Moduli. ASTM Speial Tehnial Publiation 126, 1988, pp J. Uzan, T. Sullion, C. H. Mihalek, M. Parades, and R. L. Lytton. A Miroomputer-Based Proedure for Bak-alulating Layer Moduli from FWD Data. Report FHWA/TX Texas Transportation nstitute, Texas A&M University, College Station, Sept L. W. Teller and E. C. Sutherlands. The Strutural Design of Conrete Pavemts, Part 5-An Experimtal Study of the Westergaard Analysis of Stress Conditions in Conrete Pavemt Slabs of Uniform Thikness. Publi Roads, Vol. 23, No. 8, 1943, pp F. Parker, Jr., W. R. Barker, R. C. Gunkel, and E. C. Odem. Developmt of a Strutural Design Proedure for Rigid Pavemts. U.S. Army Engineers Waterways Experimt Station, Viksburg, Miss., April P. F. Carlton and R. M. Behrmann. A Model Study of Rigid Pavemt Behavior Under Corner and Edge Loadings. Pro., 35th Annual Meeting of the Highway Researh Board, HRB, National Researh Counil, Washington, D.C., 1956, pp