Single-Event Cracking Temperature of Asphalt Pavements Directly from Bending Beam Rheometer Data

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Single-Event Cracking Temperature of Asphalt PavementsDirectly from Bending Beam Rheometer Data

Aroon Shenoy1

Abstract: Asphalt pavements under low temperature conditions develop transverse cracks as a result of one or a combinatiodistress mechanisms:~1! single-event thermal cracking;~2! thermal fatigue; and~3! load-associated thermal cracking. The crackitemperature of asphalt pavements based on the single-event thermal cracking mechanism of the asphalt binder has been suggood measure of the low temperature criterion for use as a binder purchase specification. The determination of the single-eventemperatureTcr requires a comparison of the pavement thermal stress computed from the binder stiffness data generated from thbeam rheometer~BBR! with the binder strength data measured using the direct tension test~DTT!. Thus, two instruments, each witdifferent sample preparation techniques, different sample sizes, and different data analysis methods, are required for the final dtion of a single point signifying the cracking temperature. The present work revisits the method of data analysis to determineTcr andshows that the cracking temperature can be obtained directly from the BBR data without the use of the DTT. The procedurepavement thermal stress computed from the BBR binder stiffness data to calculate the intersection of the two asymptotic therbuildup rates, which directly identifies the cracking temperature. This procedure has a significant advantage in that it makes thstand-alone device for the determination of the low temperature binder specifications.

DOI: 10.1061/~ASCE!0733-947X~2002!128:5~465!

CE Database keywords: Cracking; Temperature effects; Asphalt pavements.

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Introduction

One or a combination of three distress mechanisms—namsingle-event thermal cracking, thermal fatigue, and loassociated thermal cracking—have been attributed~Bouldin et al.2000! as the causes for the low temperature transverse crackinasphalt pavements. Single-event thermal cracking is normconsidered as the most influential of the three; hence, it wasfocus of attention when Bouldin et al.~2000! suggested a possiblapproach for the prediction of thermal cracking of pavemefrom binder properties.

Conventionally, the cracking temperatureTcr from the bendingbeam rheometer~BBR! test is determined in accordance with thAASHTO ~1999! standard test method TP1 as the higher vabetween the temperature where the stiffnessSat a loading time of60 s is 300 MPa and the temperature where them-value at aloading time of 60 s is 0.3. This Superpave criterion is sufficito minimize low-temperature cracking in all cases where unmofied binders are used. For modified binders, however, this crion was found to fail in most cases by Bouldin et al.~1999!,resulting in overprediction of performance. The direct tension~DTT! failure strain criterion based on the AASHTO~2000! stan-

1Senior Research Rheologist, Turner-Fairbank Highway ReseCenter, 6300 Georgetown Pike, McLean, VA 22101. [email protected]

Note. Discussion open until February 1, 2003. Separate discussmust be submitted for individual papers. To extend the closing dateone month, a written request must be filed with the ASCE ManagEditor. The manuscript for this paper was submitted for review and psible publication on May 9, 2001; approved on January 18, 2002. Tpaper is part of theJournal of Transportation Engineering, Vol. 128,No. 5, September 1, 2002. ©ASCE, ISSN 0733-947X/2002465–471/$8.001$.50 per page.

JOURNAL OF TRA

J. Transp. Eng. 200

,

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dard test method TP3 also failed to capture the behavior of mfied asphalts.

A mechanistic model was developed by Bouldin et al.~2000!to enable calculations of better predictive parameters fromBBR and DTT data in order to determine the relative performaof asphalt binders at low service temperatures. The dual insment method~DIM ! used the rheological data from the BBRpredict thermal stress buildup and failure data from the DTTpredict failure stress. A comparison of the thermal stress andfailure stress was then used to determine the single-event cractemperature,Tcr . This cracking temperature was suggestedBouldin et al.~2000! as the temperature to be used for the lotemperature limit in the Superpave specification, and they pposed that their suggested procedure given in the AASH~2001! draft of MP1A be used in place of the current Superpaspecification parameters.

In the present paper, it is shown that it is possible to determthe single-event cracking temperature of asphalt pavementsrectly from the bending beam rheometer data. While it is imptant to consider the thermal stress development using the Bdata, it is not necessary to superimpose the DTT failure stressto determineTcr . In fact, the single-event cracking temperatucan be very effectively determined by considering the intersecof the two asymptotic thermal stress buildup rates, whichcalculated exclusively from BBR binder stiffness data. The Dthen becomes superfluous in the method suggested by Bouet al. ~2000!. The procedure suggested in this paper has a sigcant advantage in that the BBR can be used on a stand-aloneto determine the low temperature binder specification.

Cracking Temperature Determination Procedure

The initial test temperatureTi for the BBR data is selected inaccordance with the AASHTO~1998! standard test method MP

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as one of the specification test temperatures at the 6°C incremrepresenting the low temperature binder grade. A minimumtwo different temperatures are required for deflection measments taken at various loading times of 8, 15, 30, 60, 120,240 s. The BBR test temperaturesTi andTi minus 6°C (Ti26)are selected such thatS(Ti ,60),300 MPa and S(Ti26,60).300 MPa.

The data from the BBR test conducted at various temperatfor the asphalt binder are used for the generation of the stiffnversus time curves, as shown in Fig. 1. These curves are tconverted to the thermal stress curves from which the cracktemperature can be determined. The procedure for this conveis rather complex. A qualitative outline is given below withoany equations or finer details, as these can be obtained from Bdin et al.~1999, 2000! and Dongre´ et al. ~1999!.

The shift factors for the stiffness versus time data given in F1 are determined numerically following the method given in Gdon and Shaw~1994! to produce master curves. The stiffnemaster curves are then fitted with the Christensen-Anders

Fig. 1. Example plot of stiffness versus loading time at differetemperatures obtained from bending beam rheometer measurem

Fig. 2. Example of thermal stress versus temperature curve onversion of bending beam rheometer data using TSAR Plus softw~a! Intersection using two asymptotes procedure~TAP! givesTcr ; ~b!Intersection withx-axis using single asymptote procedure~SAP!givesTcr

466 / JOURNAL OF TRANSPORTATION ENGINEERING / SEPTEMBER/OC

J. Transp. Eng. 200

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Marasteanu~CAM! rheological model~Marasteanu and Anderso1999!. The stiffness master curve is then converted to the crcompliance master curve by taking its inverse. The creep comance is further converted to the relaxation modulus througnumerical solution of the convolution integral using the methgiven by Hopkins and Hamming~1957!.

Thermal stress calculations are based on the Boltzmann’sperposition principle for linear viscoelastic materials and are cried out numerically. They involve performing three procedufor stress generation, stress relaxation, and stress summationcalculated thermally induced stresses are multiplied by the pament constant, resulting in the prediction of the thermal stresproduced in the asphalt pavements.

The above outlined procedure follows along the lines givenBouldin et al.~1999, 2000! and Dongre´ et al.~1999!. A number ofequations are used for conversion of the stiffness data fromBBR ~involving 12 or 18 data points, depending on whether tor three temperatures are used! to the final thermal stress curv~having over 25 data points!. The algorithms and calculations foarriving at the final thermal stress curve are not trivial and nethe use of software either written as a spreadsheet programcommercially available program. In the present case, the Wdows based software package TSAR Plus~Abatech, Inc.! wasused to obtain the thermal stress curve shown in Fig. 2.

It can be observed that, as the temperature lowers, the thestresses build up slowly at first and then rapidly towards theof the curve. Two asymptotes are visible, one at the beginningone at the end of the curve. The two asymptotes depict the liing rates of the thermal stress buildup. When the thermal strein the binder change from one limiting rate to the other, the trsition point identifies the temperature at which the single-evcracking would occur.@The mechanistic principle governing thicracking behavior can easily be demonstrated by holding ‘‘siputty’’ ~available in any toy shop! between the fingers of twohands and then pulling it apart at two different rates—slowlyfirst and rapidly later. The ‘‘silly-putty’’ can be extended withoubreaking when it is pulled slowly and breaks only when the ratincreased suddenly.# In the present case of the binders, the lorelaxation time will not allow the binder to respond to the suddchange in the stress buildup rate and would thus react by cracat the temperature around the transition point to relievestresses. This single-event cracking temperature,Tcr , can be de-termined from the intersection in this two asymptotes proced~TAP!, as shown in Fig. 2.

The thermal stress curve shown in Fig. 2 is constructed usa total of 25 data points starting with the temperature at whichthermal stress is the highest to the temperature where the thestress takes a value of 0.11 MPa. The two asymptotes in Fig.~a!

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Table 1. Details of Asphalt Binders Evaluated~Set 1!

Binder code Asphalt description

B6224 FluxB6225 Unmodified baseB6226 Unmodified high gradeB6227 Air-blownB6228 Modified with ElvaloyB6229 Modified with styrene-butadiene-styrene linear grafB6230 Modified with styrene-butadiene-styrene linearB6231 Modified with styrene-butadiene-styrene radial grafB6232 Modified with ethylene vinyl acetateB6233 Modified with ethylene vinyl acetate grafted

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Table 2. Details of Asphalt Binders Evaluated~Set 2!

Binder code Asphalt description

LTPP02 AC-20; Project 34-49 K-3J96-01; Kiowa County, KSPS1 and SPS9

LTPP03 AC-30 from MS State Highway Department

LTPP06 MRL 360800; GLSC, Stafford, NY, 8/16/94

LTPP07 030500 SP 55; I-90 Big Timber, MO; 850100 Pen

LTPP10 SPS1; Sample BC01; 8/30/95; 510100

LTPP11 State AL, Code: 01, SPS-5; AVC-30 12/31/91; usedvirgin test sections; VB&VW

LTPP13 SPS-5 Alberta, Canada; 10/10/90; 200/300 asphrecycled sections; 1/11

LTPP14 State code 01; Alabama SPS-5; VS-84; eastbound;12/18/91; AC-10 used in RAP mixes

LTPP15 SPS-5 Alberta, Canada; 10/3/90; 150/200 asphalt visection; can 8/11

LTPP17 SPS-5 Limon, CO; AC-20

LTPP18 MD-DT SPS-5; 24-05-00 AC for recycled sections M1992; Styrelf modified

LTPP19 SPS-5; Manitoba; Broken River Test section; 150/2Imperial Oil

LTPP20 SPS-1 AC-20; 10/18/92

LTPP21 SPS-5 Alberta, Canada; 10/3/90; 150/200 asphalt visection; can 11/11

LTPP23 240500 virgin section AC-20 SPS-5; Fredrick, MApril 1992; 11 of 11

LTPP25 Project FRI36~24! US75 019793-049; Design18-91-1887/1892; AC-5; 11 of 11

LTPP26 Mississippi State Highway AC-30

LTPP27 Mississippi State Highway asphalt

LTPP28 AC-201Rubber additive; open graded rubber mix; NSPS-5

LTPP29 MN SPS-9; Project No. TH169 SP7007-200; Grade100 Ashland refinery

LTPP30 FRI36~24! US-175; Control 0197-03-049; Desig18-91-1805/1810; AC-10w3% Latex7

LTPP31 Minn SPS-5; 85/100; sampling location 270500

LTPP33 WA SPS-8 530800

LTPP35 Maryland SPS-9 I-70 WB

LTPP36 Kansas SPS-9; 54-49, K-3196-01; Kiowa County, 9/93; NRP200902; AC-5/w polymer

LTPP37 AR SPS-9 76-22; BC02A01

LTPP38 AR SPS-1 BA-18; 050119

LTPP39 Wisconsin I-94 EB AC 85/100 antistrip added

LTPP40 MN SPS-9; Project TH169; SP7007-20; 85/100 Ko270910

LTPP42 AR SPS PG 64-22 BC51A02

LTPP43 WI SPS-9 I-94 EB, 85/100 with antistrip

LTPP44 AR PG58-72 BC 02A0 SPS-9

LTPP46 WI 550900 85/100

LTPP47 WI SPS-9; I-43 NB; and SB Amoco 85/100

LTPP48 Kansas SPS-1 and SPS-9; 54-29 K3196-01 AC-20

LTPP49 AZ SPS-9 AC-40 SHRP I; PG70-10; 048903

LTPP50 AZ SPS-9 PG 70-22; BC02A, 282, 2500

LTPP51 Young Bros 4808XX F-101; C-1321

LTPP52 WI SPS-9; WI-DOT mix; I-94 EB; 85/100 with antistri

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are determined using six data points at each end of the thestress curve, and each asymptote is mathematically expressea straight line as follows:• Asymptote 1

sTh15A1T1B1 (1)

• Asymptote 2

sTh25A2T1B2 (2)

The single-event cracking temperatureTcr is determined by theintersection of Eqs.~1! and ~2! as follows:

Tcr5B22B1

A12A2(3)

As noted earlier, the lowest thermal stress considered when fithe second asymptote was 0.11 MPa. If values of thermal strelower than 0.11 MPa are considered, then the second asymbecomes a horizontal line very close to thex-axis. In fact, in thatcase, the second asymptote can essentially be taken as thex-axis,and the intersection of the first asymptote with thex-axis wouldthen mark theTcr as shown in Fig. 2~b!. Mathematically, thecracking temperature using the single asymptote procedure~SAP!would then be given byTcr52B1 /A1 .

Experimental Details

Equipment

The bending beam rheometer from Cannon Instruments wasto get the stiffness versus loading time data at different temptures on each asphalt binder.

The direct tension test model 5525 from Instron was useddetermine the failure stress in order to make a comparisontween theTcr obtained by the dual instruments method~DIM ! ofBouldin et al.~2000! and the two asymptotes procedure~TAP! aswell as the single asymptote procedure~SAP! adopted in thepresent work.

Table 3. Comparison ofTcr Determined from Dual InstrumenMethod ~DIM ! using Bending Beam Rheometer~BBR! and DirectTension Test~DTT! Data with Those from Two Asymptotes Procedure~TAP! and Single Asymptote Procedure~SAP! using Only BBRData ~for Set 1 from Table 1!

Bindercode Tre f (°C)

Tcr (°C) usingBBR1DTT dataa

Tcr (°C) using OnlyBBR Data

Presentwork TAP

Presentwork SAP

B6224 220 235.3 234.2 233.8B6225 220 228.5 227.3 226.6B6226 220 227.1 227.2 225.8B6227 220 228.3 228.1 227.4B6228 220 234.1 232.3 231.4B6229 220 234.0 232.0 231.4B6230 220 233.3 231.2 230.5B6231 220 234.5 232.4 231.7B6232 220 231.8 230.2 229.4B6233 220 233.3 231.8 230.9aBouldin et al.~2000! DIM.

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Materials

A total of 49 binders were used in this study. They were caterized into two sets.

The first set consisted of 10 different binders that includePG52-28~flux!, a PG64-28~unmodified base!, a PG70-28~un-modified high grade!, a PG70-28~air-blown!, and six PG70-28,which consisted of the following polymer-modified systemElvaloy, styrene-butadiene-styrene linear-grafted-~SBS L-G!,

Table 4. Comparison ofTcr Determined from Dual InstrumenMethod ~DIM ! using Bending Beam Rheometer~BBR! and DirectTension Test~DTT! Data with Those from Two Asymptotes Procdure~TAP! and Single Asymptote Procedure~SAP! using Only BBRData ~for Set 2 from Table 2!


Tcr (°C) usingBBR1DTT dataa

Tcr (°C) using OnlyBBR Data

PresentworkTAP

PresentworkSAP

LTPP02 212 223.5 225.3 224.4LTPP03 212 224.7 225.9 225.4LTPP06 218 228.1 227.1 226.8LTPP07 218 228.4 227.7 227.3LTPP10 218 226.8 226.5 226.3LTPP11 212 226.8 226.8 226.2LTPP13 218 230.6 229.5 229.4LTPP14 218 231.3 231.3 230.5LTPP15 218 226.9 225.5 225.4LTPP17 220 228.4 228.0 227.2LTPP18 224 235.0 234.6 234.2LTPP19 218 233.7 232.2 231.9LTPP20 218 229.4 227.8 227.4LTPP21 218 228.2 226.1 226.0LTPP23 220 226.9 228.8 228.2LTPP25 220 230.8 230.9 230.5LTPP26 212 224.4 225.0 224.6LTPP27 212 224.5 226.5 225.9LTPP28 220 227.6 228.6 228.3LTPP29 220 232.7 233.0 232.4LTPP30 220 228.5 227.5 226.9LTPP31 212 226.0 227.2 226.3LTPP33 212 224.7 226.6 226.0LTPP35 212 226.5 225.2 224.8LTPP36 220 236.7 236.2 235.5LTPP37 212 226.6 226.6 225.6LTPP38 220 226.5 229.1 228.6LTPP39 220 226.1 226.1 225.4LTPP40 212 227.2 228.6 227.5LTPP42 220 230.1 229.9 228.8LTPP43 220 227.8 227.9 227.0LTPP44 220 231.7 231.9 231.0LTPP46 212 225.7 224.9 224.1LTPP47 212 223.9 225.8 225.1LTPP48 212 224.8 224.8 223.9LTPP49 28 217.5 218.8 217.8LTPP50 220 231.7 230.9 230.0LTPP51 212 221.0 221.8 221.0LTPP52 220 226.1 227.8 226.9aBouldin et al.~2000! DIM.


J. Transp. Eng. 200

styrene-butadiene-styrene linear~SBS L!, styrene-butadienestyrene radial-grafted~SBS R-G!, ethylene-vinyl acetate-~EVA!,and ethylene-vinyl-acetate grafted~EVA G!. All these unmodifiedand modified asphalts, which are given in Table 1, are part ofextensive ongoing polymer research program being carried othe Pavement Testing Facility located at the Turner-FairbHighway Research Center.

The second set, consisting of 39 binders, also includedmodified and modified binders, but these were the ones that wused in the Long Term Pavement Performance~LTPP! Program.The details of these binders are given in Table 2.

Results and DiscussionThe entire operation of applying the asymptotes was dstraightforwardly on an Excel spreadsheet. The straight lineeach asymptote was found to fit in most cases with anR2 between0.970 and 0.999. A comparison of the single-event cracking tperature by the DIM~using the BBR and DTT data! and by theTAP and SAP suggested in this work~using only BBR data! isshown in Table 3 for Set 1 of the 10 asphalt binders and in Ta4 for Set 2 of the 39 asphalt binders. The data cover a wide sof cracking temperatures from217 to 235°C. This range wouldencompass most unmodified and modified asphalt binders thanormally in use.

The maximum difference between theTcr obtained by theDIM was found to be about 2°C for TAP and about 2.8°C for SAin four out of the 49 asphalt binders analyzed. In the majoritythe cases, the difference between theTcr obtained by the DIMwas less than 1°C for TAP and less than 1.5°C for SAP. In fawhen a comparison of theTcr obtained by the DIM is made withTAP and SAP, as shown in Figs. 3 and 4, the best line throughcomparison fits with anR2 of 0.9 for both TAP and SAP, therebshowing an excellent correlation.

It was seen that, wherever there was a difference betweenTcr obtained by the DIM and the present work, the TAP and Sunderpredicted in most cases. Thus, TAP and SAP would be mconservative than the DIM in their predictions of the crackitemperature. Since theTcr obtained by the DIM of Bouldin et al.~2000! can by no means be considered a ‘‘gold’’ standard, therno way of telling which of the methods, the DIM or the TAP/SAis closer to reality unless a comparison is made with actual fiperformance data.

Limited field performance data are available~EBA 1994;Gavin and Dunn 1997; Anderson et al. 1999; Bouldin et al. 192000; Dongre´ et al. 1999! on the Lamont test road built in 1991 iAlberta, Canada. Sections of this road were constructed wseven different conventional and air-blown binders uniquely csen to purposefully exhibit very good and very poor low tempeture performance. One binder, described in Table 5, whichused in Lamont test road section 1, is chosen here to checkpredictions of the TAP and SAP in estimating the cracking teperature. The temperature range where cracking occurred infield was between230.2 and233°C. The DIM using BBR andDTT data gives a value of235.4°C, which is an overpredictionFig. 5 shows that theTcr obtained by TAP is233.5°C while thatby SAP is233.2°C. These values are closer to reality and inway certify the efficacy of the suggested procedure.

In the discussion so far, the reference temperatureTre f usedfor shifting the data in forming the master curve has not bealluded, though these have been given in Tables 3 and 4. A chin the reference temperature makes a difference in the positiothe thermal stress curve, thereby resulting in a different value

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Fig. 3. Comparison ofTcr determined from bending beam rheometer~BBR! and direct tension test~DTT! data by dual instrument method~DIM !with Tcr determined directly from BBR data by two asymptotes procedure~TAP!

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theTcr , and this point must be borne in mind. For example, incase of the Lamont sample, the BBR data sets were availabsix different temperatures of230, 227, 225, 224, 221, and218°C. When the reference temperature is changed from224 to227°C for the same six sets of data, theTcr value from the DIMusing BBR and DTT data can go from235.4 to 236.3°C, asshown in Table 6. This difference may appear small, but conering that an attempt is being made to do a field validation whthe cracking occurred between230.2 and233°C, the overpre-diction goes to a higher level if the comparison is made w236.3 instead of235.4°C.

The reference temperatures for the binders in Table 4 areferent in many cases; therefore, a comparison between theTcr

values may not be the true indicator of their expected permance. On the other hand, in Table 3, the reference temperfor each binder is the same (Tre f5220°C) and hence theTcr

values for these binders can be used for comparing the expeperformance of these binders. However, theTre f5220°C is ar-bitrarily chosen and not related to any property of the bindercomes into the calculation because of the use of an Arrhenius

JOURNAL OF TRA

J. Transp. Eng. 200

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of equation for the shift factor. The temperature dependencestiffness or viscosity is normally described by the Arrhenius eqtion at temperaturesT.Tg1100 wherein the forces of intermolecular interactions need to be overcome~Shenoy 1999!. On theother hand, at temperaturesT,Tg1100, the temperature dependence of stiffness or viscosity is best described by the WilliamLandel-Ferry ~W-L-F! equation, since free volume and ichanges with temperature play a dominant role at temperatrelatively near theTg ~Shenoy 1999!. In the analysis of BBR dataone would therefore expect that the W-L-F equation shouldused in preference to the Arrhenius type equation. If this is dothen the Tre f could be linked to the Tg by choosingTre f5Tg1~say, 10 or 20!, thus getting a meaningful definitionthat is related to the property of the binder. This was not donethe present work, as the focus was not on how to get a standized thermal stress curve but rather on showing howTcr can beobtained directly from it. However, if a standardization of theTcr

is done in the future, then it might be worth considering the usethe W-L-F equation for the shift factors in place of the Arrhenitype equation that is presently being used.

Fig. 4. Comparison ofTcr determined from bending beam rheometer~BBR! and direct tension test~DTT! data by dual instrument method~DIM !with Tcr determined directly from BBR data by single asymptote procedure~SAP!


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Fig. 5. Determination ofTcr directly from bending beam rheomete~BBR! data by~a! two asymptotes procedure~TAP! and ~b! singleasymptote procedure~SAP! for asphalt binder used in Lamont teroad section 1

Table 5. Details of Asphalt Binder used in Lamont Test RoSection 1

Parameter Value

Grade, type 80/100B, air blownSupplier, location Esso, Port Moody~British Columbia!Penetration; units 25°C 100100 g, 5 s, 10°C 22100 g, 5 s, 5°C 13Viscosity, Pa•s, 60°C 96mm2/s 277

Note: This asphalt binder designated as 80/100 Air Blown was choserepresent a low viscosity, highly temperature susceptible crude sothat was air blown to improve its temperature susceptibility.

Table 6. Comparison ofTcr Determined from Duel InstrumenMethod ~DIM ! using Bending Beam Rheometer~BBR! and DirectTension Test~DTT! Data When Different Reference Temperaturesused for Shifting BBR Curves


Tcr (°C) usingBBR1DTT Dataa

Tcr (°C) using Only BBRData

Present workTAP

Present worSAP

LamontIsxI1 230 235.9 234.3 234.0227 236.3 235.1 234.7225 235.5 233.5 233.2224 235.4 233.5 233.2221 235.6 233.7 233.3218 235.5 233.7 233.2

aBouldin et al.~2000! DIM.


J. Transp. Eng. 200

Acknowledgments

The writer wishes to express his gratitude to Susan Needhamgenerating the BBR and DTT data used in this paper, and alsouseful discussions on the subject matter. The writer is also grful to Ernest J. Bastian, Jr., for his valuable comments.

Notation

The following symbols are used in this paper:A1,B1,A2,B2 5 coefficients in Eqs.~1! and ~2!;

m 5 slope of stiffness versus loading time curveat 60 s;

S 5 stiffness modulus~MPa!;T 5 temperature~°C!;

Tcr 5 single-event cracking temperature~°C!;Ti 5 initial test temperature~°C! chosen for BBR

measurement;Tre f 5 reference temperature~°C!;

t 5 loading time~s!; andsTh 5 thermal stress~MPa!.

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Documents

Single-Event Cracking Temperature of Asphalt Pavements Directly from Bending Beam Rheometer Data