BOUSDEF: A Backcalculation Program For Determining Moduli .

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166TRANSPORTATION RESEARCH RECORD 1260BOUSDEF: A Backcalculation Program forDetermining Moduli of a PavementStructureHAIPING ZHOU,R. G.HICKS, ANDc.A.Highway and transportation agencies have an increasing responsibility for the maintenance, rehabilitation, and management ofhighways, particularly with regard to asphaltic concrete pavements. Efficient and economical methods are required for determining the structural properties of existing flexible pavements.Nondestructive testing (NDT) of pavements is one of the mostuseful and cost-effective methods for evaluating the structuraladequacy of pavements. With the wide use of NDT, in particularthe deflection test, a large amount of test data can be obtained.One common use of deflection data is to determine the pavementlayer moduli through backcalculation. The microcomputer program BOUSDEF for backcalculating the moduli of a pavementstructure using deflection basin data is presented. The solutiontechniques for use in developing the program are described,including the use of the method of equivalent thicknesses, Boussinesq theory, consideration of nonlinearity of pavement materials, and consideration of overburden pressure on stress calculation. Evaluation of the program was performed by twoapproaches: (a) comparing the backcalculated moduli with theoretical moduli, and (b) comparing the backcalculated moduliwith results from other developed backcalculation programs. Theevaluation shows that the moduli backcalculated using the BOUSDEF program compare well with the theoretical moduli and alsoare compatible with those from other developed programs. TheBOUSDEF program runs fast compared with other backcalculation programs; therefore, the program can be effectively usedas a tool to make initial evaluations of deflection testing data fordetermining pavement layer moduli.Highway and transportation agencies have increasing responsibility for maintenance, rehabilitation, and management ofhighways, particularly with regard to asphaltic concrete (AC)pavements. Efficient and economical methods are requiredfor determining structural properties of existing flexiblepavements.Pavement structural properties may be generally stated interms of the resilient modulus, which is a key element inmechanistic pavement analysis and evaluation procedures. Fora multilayer pavement structure, the resilient modulus of eachpavement layer may be determined by two possible methods-destructive testing and nondestructive testing (NDT).Destructive testing is generally done by obtaining cores froman existing pavement and testing them using laboratory equip-H. Zhou, Oregon Department of Transportation, Highway MaterialsLaboratory, 800 Airport Road, S.E., Salem, Oreg. 97310. R. G.Hicks and C. A. Bell, Department of Civil Engineering, Oregon StateUniversity, Corvallis, Oreg. 97331.BELLment. NDT, on the other hand, uses deflection basin datagenerated from an NDT device to quantify the response of apavement structure due to a known load. The known responseis then used in a backcalculation procedure, which generallymeans using the deflection basin data to determine the pavement layer moduli. The NDT method has certain advantagesover the destructive method, such as no physical damage tothe pavement structure, and requiring no laboratory tests.NDT of AC pavements is one of the most useful and costeffective methods developed by engineers to assist in themanagement of pavements. With the increased responsibilitythat highway agencies have for effectively apportioning fundsand efficiently designing major rehabilitation projects, theuse ofNDT methods has become, or in some cases, can become,an invaluable aid in determining the actual condition of pavement sections in a highway network (J). The emphasis inthe 1986 AASHTO Guide for Design of Pavement Structures(2) on use of the resilient moduli of pavement materials inpavement design and on use of NDT in overlay design alsosuggests that these methods will have increased usage in thefuture.The analysis of NDT data to determine pavement layerproperties requires use of mechanistic methods. The principalobjective of mechanistic analysis of NDT data is to producemoduli of pavement layers for in-service temperatures at various load levels. These mechanistic methods assume thatstresses, strains, and deformations in pavements can be modeled as multilayered linear or nonlinear elastic structures,resting on linear or nonlinear elastic foundations, as shownin Figure 1. This capability makes it possible to use a trialand-error procedure to assume the layer properties, calculatethe surface deflections, compare these with the measureddeflections, and repeat the procedure until the calculated andmeasured deflections are acceptably close. Several such backcalculation methods of analysis have been developed usingdifferent assumptions or algorithms concerning the layermaterial properties, all of which have the trial-and-error procedure as their basis. One drawback of all the available programs is computing efficiency, which seriously impacts theiruse in routine design work.BOUSDEF is a much faster backcalculation program. Theprogram is based on the method of equivalent thicknesses andmodified Boussinesq equations. The solution technique,development of the program, and comparison with otherbackcalculation programs are described in the followingsections.

Zhou et al.167pFIGURE 1 Generalized rnultilayered elastic system.SOLUTION TECHNIQUEEmodulus of elasticity, andPoisson's ratio.µThe BOUSDEF program includes the following techniques:1. Use of the method of equivalent thicknesses,2. Use of Boussinesq theory,3. Consideration of nonlinearity of pavement materials, and4. Consideration of overburden pressure .The following paragraphs briefly describe these techniques .For a two-layer system, the equivalent thickness of a layerwith modulus 2 and Poisson's ratio µ 2 relative to a layer ofthickness h 1 , modulus 1 , and Poisson's ratio µ 1 , may beexpressed by equating the stiffness of both layers, that is,or ,Method of Equivalent ThicknessesThe method of equivalent thicknesses (3) assumes that anytwo layers with similar structural stiffness will distribute loading in the same way. According to this assumption, all layersin a multilayered structure can be converted to one layer withequivalent stiffness by using the following relationship:(1)12(1 -(2)µDRearranging the equation,h h2DµD12(1 -I [E1 (1 - µDE2 (1 - µf) ]1/3By expanding this concept for a multilayer system as shownin Figure 2, a general form of the equation may be writtenwheren- 1Dhstiffness,layer thickness,h,;2:i IE, (1 - f.l )]uh; [E" 1 - µf)3(3)

168TRANSPORTA TlON RESEARCH RECORD 1260FIGURE 2 Conceptual representation of method of equivalent thicknesses.whereFor a uniformly distributed load (Figure 3b), integration ofEquation 4 yieldsh.;equivalent thickness for ith layer,thickness of ith layer,E;modulus of ith layer,E"modulus of nth layer,µ; Poisson's ratio for ith layer, andµ,, Poisson's ratio for nth layer.h;d (1, µ)cr 0aE· [[l ( /z)j l2 (1 -22µ) { [1 (zla) ]112- } J(5)whereLimitations of the Method of Equivalent ThicknessesThere are a number of limitations with regard to the use ofthe method of equivalent thicknesses. One is that the pavement layer moduli should decrease with depth, preferably bya factor of at least two between consecutive layers . Anotheris that the equivalent thickness of a layer should preferablybe larger than the radius of the loaded area (4).Boussinesq Equations for DeflectionsWith the use of the equivalent thicknesses method, the Boussinesq equation for calculating deflection at a depth z andradius r in an elastic half-space can be applied to a multilayerelastic system (3). The general equation for deflection due toa point load , as shown in Figure 3a, is,(4)whered, , deflection at depth z and radius r,P point load,R distance from point load to the location where deformation occurs,E modulus of elasticity, ande angle between centerline of load and location ofanalysis (see Pigure 3a).d,a0deflection on the load axis,stress under the loading plate,aradius of the loading plate, andz - depth where deform ation occurs.Equation 5 for the uniformly distributed load is valid onlyfor calculation of deflections on the load axis. For points offthe axis of the load, the integration cannot be carried outanaiyticaliy, but for iayered systems with a stiff top iayer,Boussinesq's equation for a point load, Equation 4, will usually give satisfactory results (3) .Boussinesq Equations for StressesBoussinesq also formulated equations for calculating stressesfor a homogeneous, isotropic, linear, elastic semi-infinite space.The use of the method of equivalent thicknesses allows theseequations to be used for a multilayer pavement system. Fora load uniformly distributed over a certain area as shown inFigure 3b, the normal stresses can be determined using thefollowing equations:(6)r1 2µ CToi - 'J- - r1l"""L L1 µfnl )2ll12J-LI\""1"'1. "fl ra/\ .}2jJl2(7)

Zhou et al.169pa) Point Loadpb) Distributed LoadFIGURE 3condition.Conceptual representation of Boussinesq's half-space loadingwhereu z vertical stress, andu, u, horizontal stresses.These equations will be used to calculate stresses inducedby loadings.Correction Factors for Boussinesq MethodThe use of the method of equivalent thicknesses allows theBoussinesq theory to be applied in a multilayer system. Stresses,strains, and deformation at any point in an elastic halt-spacecan be determined by using corresponding Boussinesq equations. In order to obtain good agreement between the stresses,

170TRANSPORTATION RESEARCH RECORD 1260strains, and deflection calculated by the Boussinesq approachand by exact elastic theory, Ullidtz and Peattie (3) suggestthat correction factors should be applied to the equivalentthicknesses. For the simple case of calculations on the axisof a uniformly distributed load, Equation 3 is modified asfollows:fh;, Ln- I. 1h,[E, (l - µ En (1 - µ;)' ) ] l/3(8)where f is a correction factor; for a two-layer system ,f 0. 9;for a multilayer system ( 2 layers) ,f LO for the first layer,0.8 for the rest of the layers.Additional correction factors are required when usingEquation 4 for the point load for more general analysis ofdeflection, because the assumption that the uniformly distributed load can be approximated by a point load producesinaccuracies near the surface of the pavement. These corrections are as follows (5):Z'' z;a2 Z, 0.6 Z,J.Sa2(1 - µ;) - [2(J - µ;) - 0.7](Z/ 2a)Z,2:Z, aa(9a)(9b)wherez;Z, a corrected equivalent thickness for ith layer,h;,, modified equivalent thickness for ith layer, andload radius.thicknesses by their respective densities and summing theseto the desired depth . The total vertical stress u,., is the sumof the load-induced stress uv 1 and overburden pressure ,IIUvt 2: h/'/;(11)i lwhereh, thickness of ith layer, and-y, density of ith layer.The total horizontal stress u 1,, is a function of the loadinduced horizontal stress u hi plus horizontal stress due to overh11rNPnnrPCCll"f'P. . .t'. "4 . , uh, Uhl KoL" h;'Y;(12)i lwhere K 0 is the coefficient of at-rest earth pressure.These expressions do not include a term for pore water pressure, because pore water pressure is a function of groundwater table depth. The assumption is made that the groundwater table is at depth below the top of the subgrade andtherefore does not affect the results.The coefficient of at-rest earth pressure K 0 is a function ofthe angle of friction P for a given soil as determined by atriaxial compression test. For granular soils,K 0 1 - sin P(13a)and for fine-grained soils (6),Consideration of Nonlinearity of Lower LayerMaterialsK 0 0.95 - sin PThe resilient properties of pavement materials, specially thosecoarse grained and fine grained, are generally stress dependent. The resilient moduli of the these materials varyaccording to the stress state within the layers. The moduli ofthese materials are usually approximated by the followingrelationships:Das (7) reported an approximate range of P from 25 to 38degrees for normally consolidated clays and from 26 to 46degrees for sands. Overall, this represents a range of K 0 from0.28 to 0.56. For most geotechnical work, when triaxialcompression test data are not available, a value of 0.5 isassumed for K0 ( ).for coarse-grained materials, or(lOa)for fine-grained materials.(!Ob)whereMR resilient modulus (psi),e bulk stresses (psi),ud deviator stress (psi), andk,, k 2 regression coefficients that depend on materialsproperties.Most often, these coefficients are determined through laboratory tests.Consideration of Overburden StressesActual stresses in a pavement structure consist of two partsload-induced and overburden stresses. For vertical stresses,il11:: ove1 bu1de1i p1essure is calculated by multiplying the layer(13b)DEVELOPMENT OF THE BOUSDEF COMPUTERPROGRAMProgram FlowchartThe BOUSDEF program is developed for determining in situmoduli of a pavement structure using deflection data througha backcalculation technique. Figure 4 shows a flow diagramof the program.To start with, the program first reads input data sets thatinclude NDT load force and load radius, pavement layerthicknesses, Poisson's ratio, minimum, maximum, and initialmodulus, density of pavement materials , deflection data (upto seven sensor readings), percent tolerance to stop the deflection matching process, and number of iterations. By callingthe subroutine DEFLECTION, which uses the solution techniques described earlier, the initial modulus and layer thickness information are used to determme the equivalent thick-

Input data setsLoad level i 1Call DEFLECTIONCompute sum of error from% difference of deflectionsCompute baselinedeflectionCompute for eachlayer an E-AltNoE-Alt EmaxCall DEFLECTIONE-Alt EminCompute S & A matricesConstruct Band Cmatrices directlyNoBased on least square summationand weighting factor constructB & C matricesFIGURE 4 Flowchart of BOUSDEF program (co11tinued on next page).Compute deflectionfor changed E-Alt

Solve set of simultaneousequations for computed E'sCallSIMEQU{B}{E} {C}Reconstruct B & Cmatrices to set E'sequal limitsSet computed E's asnew baseline E'sCompute new baselinedeflectionsCall DEFLECTIONCompute sum of percent errorNoi i 1Calculate stresses undereach load levelCalculate stresses underdead loadCalculate bulk stressesCalculate deviate stressesDetermine k1 & k2 for bothbase and subgradePrint summary resultsFIGURE 4 (continued)

173Zhou et al.3. Coefficients k, and k 2 for base and subgrade layers,appearing in Equations lOa and lOb.Calculated from layeredr tlcprng amMeasuredDeflection 1-- ., ,.:.,.,Def A S*Log(E)c:0 :;cE(min.)E(est.)E(max.)Log ModulusExampleAn example is provided to illustrate the use of the program.Table 1 presents the pavement and deflection test data forthe example. The pavement is a conventional flexible structure with 8-in . asphalt concrete surface , 12-in. aggregate base,and infinite depth of subgrade. Deflection testing was performed using a falling weight deflectometer (FWD) on oneshort section of a road.By using the BOUSDEF program, resilient modulus foreach pavement layer was determined and presented in Table2. Bulk stresses in the middle of the base layer and deviatorstresses on the top of subgrade are calculated. Regressioncoefficients k 1 and k 2 for both base and subgrade are alsodetermined. As can be seen in Table 2, both base and subgradematerials appear to have a nonlinear property with k 2 0.58for base and - 0.13 for subgrade. The results are plotted inFigure 6.FIGURE 5 Simplified description of deflection matchingprocedure.Sensitivity to the User Inputnesses. Deflections for the given NDT load and load radiusare then calculated . The calculated deflections are comparedto measured deflections. If the sum of the differences is greaterthan the tolerance specified by the user, the program will startiterations by changing the moduli to compute a new set ofdeflections.A simplified description of the deflection matching procedure is shown in Figure 5. This process repeats until the sumof the differences is less than the tolerance or the maximumnumber of iterations has been reached. This procedure isrepeated for each load level until all deflection data are used.The moduli determined from each set of deflection basindata are used to calculate normal stresses induced by load.Stresses under the deadload of the upper pavement materialsare also determined. For the base layer, bulk stresses in themiddle of the layer are calculated . For the subgrade, deviatorstresses on the top of subgrade are determined. These stressvalues and moduli are then regressed to find coefficients k 1and k 2 for both base layer and subgrade.The backcalculated modulus corresponds to an averagecondition in the pavement material, whereas the bulk anddeviator stresses are calculated under the load at the middleof the base layer and the top of the subgrade rather thanthrough the entire body of the base and subgrade. Therefore,the nonlinear analysis is limited to the stress condition at aspecific location rather than at different depths of base andsubgrade. Also, the method of equivalent thicknesses orBoussinesq approach is least reliable in predicting horizontalstresses (3).Program OutputThe program has the capability of determining the following:1. Resilient modulus for each pavement layer.2. Bulk stresses and deviator stresses induced by both loadand deadload of upper-layer pavement materials .The initial moduli specified by the user seem to have minoreffect on the final backcalculated moduli . This featureTABLE 1 PAVEMENT AND DEFLECTION DATA FORTHE EXAMPLEPavement DataTh ic kn ess Poiss on' s ra t ioLaver8"0. 35ACAgg. Base12"0. 400. 40SubgradeDens ity (pcf )144120100Deflection Data36"58"8"18"Load Sensor O"Deflection Readings (mils)(lbs)6.07 4. 04 2.41 1. 25 0.9127896.59 4.02 2.41 1. 37 0.9430356.55 3.89 2.28 1. 50 0.943055652112 . 92 8.26 6.47 3 . 19 1. 82664413 .18 8.81 7.23 3. 53 1. 82656213.82 9.57 6.47 3.88 1. 72652113.31 8. 26 7.10 3.53 1. 94648013.05 8.48 5.58 3.65 1. 93648013 . 44 12 . 72 7.48 5. 59 3. 501144222.09 14.35 11. 92 5.81 3.7622.48 15 . 44 13 .19 6.38 3.96117701160623. 77 16.74 11. 79 6.84 3.8322.99 14.78 12.68 6.84 3.97114421177022 . 35 14 . 78 10 . 65 6.84 3. 91Note: Load radius is 5. 9 inches

174TRANSPORTATION RESEARCH RECORD 1260TABLE 2 SUMMARY OF BACKCALCULATION RESULTSFOR THE EXAMPLESummary of Non-linear Characteristics of Lower LayersFor base layer : kl kl For subgrade:806918687k2 0.58k2 -0 .13Summary of Moduli and Stresses *Load (lb)E(l)2,789 106,43283,3623,0353,05574,9786,480 104,0876,480 399,3596,521 117,9826,52199,3146,562 142,5816,644 158,74011, 442 117,18011,442 100,93911,606 136,67311, 770 156,59911,770 105,657Average 135,994E(2)26,91138' 269' 77335' 13541,68069,787E(3)16,37716,87016 , 13,53313,37613' 7 .8331. 3523.6124 .1831. 79DST RS5.595.765. 597. 755.968.017. 448.407.9610.559.6511.1610 . 46ID .1814,32642,6891,000 . . . . . . 110100Bulk Stress (psi)100,000 .- - ----,---. - -- - -, - - --- - --- -. ----- -- ----- ,- - ------ --,- --,- - - - -. L. 1 . ,,, . , . .1 --- -.1 - .-··- - ----- ------ ---···· ·-·--··i-------{--·--·-- - . r- - - - - . - . - - - - - ' - . - - - - - - - - - - - - -. - - - - - - . , . - - - - - -- -- - - --f- - - - - - - - - - - - --"iii.e:-(/).ft. .::J 10,000f.- - - - · - ' - - . . :tI- - . . -- - - - - - .- - :ic: "iii ---- ---l--- ---- -- -- ---1--- ----·- : -··-- -- -- . --- - r--·-··--l-. ·- -. - -:- --- .

equivalent stiffness by using the following relationship: D where D h (1) stiffness, layer thickness, E µ modulus of elasticity, and Poisson's ratio. 167 For a two-layer system, the equivalent thickness of a layer with modulus 2 and Poisson's ratio µ2 relative to a layer of thickness h 1 , modulus 1, and Poisson's ratio µ may be

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