This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles

for the
Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.

Designation: D8458 − 22

Standard Test Method for

Evaluation of Fatigue Performance of Asphalt Mixtures
Using the Three-Point Bending Cylinder (3PBC) Test1
This standard is issued under the fixed designation D8458; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.

1. Scope 2. Referenced Documents

1.1 This test method provides a procedure to determine the 2.1 ASTM Standards:2
fatigue life (number of cycles to failure, Nf) of asphalt D8 Terminology Relating to Materials for Roads and Pave-
mixtures, and also the reduction in dynamic modulus (|E*|) ments
with loading cycles, using cylindrical samples subjected to D979/D979M Practice for Sampling Asphalt Mixtures
three-point cyclic bending. The results obtained from this test D3549/D3549M Test Method for Thickness or Height of
can be used to calibrate Viscoelastic Continuum Damage Compacted Asphalt Mixture Specimens
(VECD) models to obtain a damage characteristic curve, which D3666 Specification for Minimum Requirements for Agen-
in turn can be used to obtain fatigue lives (Nf) at a variety of cies Testing and Inspecting Road and Paving Materials
temperatures, strain levels, and frequencies (a separate stan- D5361/D5361M Practice for Sampling Compacted Asphalt
dard practice is being drafted for this procedure). Even though Mixtures for Laboratory Testing
this test method is intended primarily for displacement (strain) D6373 Specification for Performance-Graded Asphalt
controlled fatigue testing, certain sections may provide useful Binder
information for force-controlled tests. 2.2 AASHTO Standards:3
1.2 The test method describes the testing apparatus, R 30 Practice for Mixture Conditioning of Hot Mix Asphalt
instrumentation, specimen fabrication, and analysis procedures (HMA)
required to determine the number of cycles to failure of asphalt R 83 Preparation of Cylindrical Performance Test Speci-
concrete. mens Using the Superpave Gyratory Compactor (SGC)
1.3 The text of this test method references notes and T 378 Standard Method of Test for Determining the Dy-
footnotes which provide explanatory material. These notes and namic Modulus and Flow Number for Asphalt Mixtures
footnotes (excluding those in tables and figures) shall not be Using the Asphalt Mixture Performance Tester (AMPT)
considered as requirements of the test method. M 320 Standard Specification for Performance-Graded As-
phalt Binder
1.4 Units—The values stated in SI units are to be regarded M 332 Standard Specification for Performance-Graded As-
as the standard. No other units of measurement are included in phalt Binder Using Multiple Stress Creep Recovery
this standard. (MSCR) Test
1.5 This standard does not purport to address all of the
safety concerns, if any, associated with its use. It is the 3. Terminology
responsibility of the user of this standard to establish appro- 3.1 Definitions:
priate safety, health, and environmental practices and deter- 3.1.1 dynamic modulus, |E*|—the amplitude of the complex
mine the applicability of regulatory limitations prior to use. modulus that defines the relationship between the stress and
1.6 This international standard was developed in accor- strain of viscoelastic materials. The |E*| is simply the peak-to-
dance with internationally recognized principles on standard- peak stress divided by peak-to-peak strain in a cyclic test run
ization established in the Decision on Principles for the at a constant frequency.
Development of International Standards, Guides and Recom-
mendations issued by the World Trade Organization Technical
Barriers to Trade (TBT) Committee. 2
For referenced ASTM standards, visit the ASTM website, www.astm.org, or
contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM
1
This test method is under the jurisdiction of ASTM Committee D04 on Road Standards volume information, refer to the standard’s Document Summary page on
and Paving Materials and is the direct responsibility of Subcommittee D04.26 on the ASTM website.
3
Fundamental/Mechanistic Tests. Available from American Association of State Highway and Transportation
Current edition approved Dec. 15, 2022. Published January 2023. DOI: 10.1520/ Officials (AASHTO), 444 N. Capitol St., NW, Suite 249, Washington, DC 20001,
D8458-22. http://www.transportation.org.

Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States

1
 D8458 − 22
3.1.2 initial dynamic modulus, |E*|0—initial undamaged results. Reliable results may depend on many factors; following the
dynamic modulus determined at approximately 50 load cycles. suggestions of Specification D3666 or some similar acceptable guideline
provides a means of evaluating and controlling some of those factors.
3.1.3 Poisson’s ratio—the ratio of transverse to longitudinal
strains of a loaded specimen. 6. Apparatus
3.1.4 Timoshenko beam theory—the theory which considers
6.1 Test System—The test system consists of a three-point
the bending and shear effects when subjected to loading, and
bending cylinder test setup, a load frame capable of providing
commonly used in low aspect ratio beams (that is, when
length-to-diameter ratio of a beam is <6). cyclic load and displacement, an environmental chamber
(temperature control system), and closed-loop control and data
3.2 For definitions of other terms used in this standard, refer acquisition system. Fig. 1 illustrates the details of the three-
to Terminology D8. point bending cylinder setup. The side clamps shall hold the
asphalt sample fixed with no sliding, rotation, or combination
4. Summary of Test Method
thereof. High-strength steel shall be used in the production of
4.1 A cylindrical specimen is clamped using C-shaped the fixture. The test system’s minimum requirements are
clamps in a three-point bending setup and subjected to sinu- specified in Table 1.
soidal actuator displacement-controlled loading with zero
mean. While the loading is applied at the central clamp, the 6.2 Three-Point Bending Cylinder (3PBC) Test Fixture—
side clamps hold the sample in a fixed position. The actuator The test fixture is composed of a 175 mm solid base, two fixed
displacement is set such that the initial strain ranges from 200 95 mm end supports used to clamp the sample, and a 95 mm
to 800 × 10–6 mm/mm. The recommended loading frequency central clamp for application of cyclic (zero-mean) vertical
rate is from 5 Hz to 10 Hz. The load and deflection at the load. Supports and loading clamps are composed of two
central clamp are measured during the entire duration of the C-shaped pieces, which are screwed together to hold the
test to be later analyzed by Timoshenko beam theory formu- asphalt sample in place. The lower C-shaped pieces are welded
lations to calculate the change in dynamic modulus (|E*|) per to the base plate. The distance between two supports is 125 mm
loading cycle. and the inner diameters of clamps are 68 mm each. Top bars are
placed on side clamps to prevent any potential deflection of the
5. Significance and Use side clamps (see Fig. 1). All the parts (except the LVDT
holders) are made of high-strength steel to prevent any
5.1 This test method can be utilized to determine the fatigue
undesirable deformation during the test. It is noted that
resistance of asphalt mixtures. The test method is generally
dimensional tolerances shown in Fig. 1 are mandatory.
valid for specimens that are tested at intermediate tempera-
tures. The three-point bending cylinder test samples are ob- 6.3 Loading Device—The test system includes a closed-
tained by coring a 68 mm diameter cylinder from the center of loop, computer-controlled loading component which, during
a 150 mm diameter gyratory compacted sample, or horizontal each load cycle in response to commands from the data
coring from field cores or slabs cut from field sections. After processing and control component, adjusts and applies a load
coring, the sample is ready for testing and no further sample such that the specimen experiences a constant level of displace-
preparations steps are required. The two ends of the 68 mm ment during each loading cycle. The loading device should be
diameter three-point bending cylinder sample do not need to be capable of providing peak-peak sinusoidal loading with zero
sliced. mean at a frequency range of 5 Hz to 10 Hz. Fig. 2(a) and Fig.
5.2 The Timoshenko beam theory is used to calculate the 2(b) show the three-point bending cylinder setup with a
reduction in dynamic modulus for each loading cycle. The test mounted specimen in a material testing system and asphalt
can be used to investigate the fatigue behavior of asphalt mixture performance tester, respectively.
mixtures at various strain levels, temperatures, and frequen- 6.4 Environmental Chamber (Temperature Control
cies. The results can be used to compare the fatigue life (Nf) for System)—The environmental chamber shall enclose the entire
different asphalt mixtures. The Nf value can be calculated as specimen and the fixture and maintain the specimen at the
the 50 % reduction in dynamic modulus. The Nf value is an desired test temperature within 60.5 °C throughout the condi-
indicator of fatigue performance of asphalt mixtures containing tioning and testing times. An environmental chamber is not
various mix design properties, asphalt binder types and required if the temperature of the surrounding environment can
modifications, gradations, and recycled materials. Typically, a be maintained within the specified limits.
higher Nf value indicates better fatigue performance. The Nf 6.4.1 Control and Data Acquisition System—During each
value may be used to identify crack-prone mixtures in load cycle, the control and data acquisition system shall be
performance-based mix design or in construction acceptance capable of measuring the displacement of the beam specimen,
procedures, or both. and adjusting the load applied by the loading device such that
NOTE 1—The quality of the results produced by this test method are the specimen experiences a constant level of displacement on
dependent on the competence of the personnel performing the procedure each load cycle. In addition, it shall be capable of recording
and the capability, calibration, and maintenance of the equipment used. load cycles, applied loads, beam displacements, and tempera-
Agencies that meet the criteria of Specification D3666 are generally
considered capable of competent and objective testing, sampling, ture while computing and recording the maximum tensile
inspection, etc. Users of this test method are cautioned that compliance stress, maximum tensile strain, phase angle, and dynamic
with Specification D3666 alone does not completely ensure reliable modulus at load cycle intervals specified by the user.

2
 D8458 − 22

FIG. 1 General Schematic View of the Three-Point Bending Cylinder Test Setup with a Loaded Specimen: (a) Elevation View; (b) Side
View; and (c) Plan View

6.5 Deformation Measurements—Mid-span deformation same LVDT type can be used to perform dynamic modulus
shall be measured using sensors mounted on two sides of the tests according to AASHTO T 378.
central clamp as shown in Fig. 2. Also, it is encouraged to use
a lateral LVDT to monitor the lateral movement of the side 6.6 Caliper or ruler accurate to 60.05 mm for specimen
clamps, if applicable. Spring-loaded linear variable differential diameter measurement.
transducers (LVDTs) are recommended but not specified. The

3
 D8458 − 22

FIG. 1 General Schematic View of the Three-Point Bending Cylinder Test Setup with a Loaded Specimen: (a) Elevation View; (b) Side
View; and (c) Plan View (continued)

6.7 The temperature measurements shall be performed us- 8.2 If the testing is run for the purpose of ranking multiple
ing a calibrated digital thermometer with a tolerance range of asphalt mixtures, a minimum of two replicates per testing
no more than 60.2 °C. condition is recommended. If a complete fatigue curve is
6.8 Data Quality—Accept only test data meeting the data needed for use in the viscoelastic continuum damage (VECD)
quality statistics given in Table 2. Calculation steps and all the analysis, a minimum of four replicates are recommended,
formulations for these data quality statistics are provided in where two replicates will be tested at one temperature and the
Annex A2. other two will be tested at another temperature. Otherwise,
prepare as many samples as required.
7. Hazards
8.3 Laboratory-Mixed and Laboratory Compacted (LMLC)
7.1 Observe standard laboratory safety precautions when Specimens—The 3PBC specimen shall have a diameter of 68 6
preparing and testing asphalt concrete specimens. 0.5 mm and a minimum length of 150 mm. LMLC specimens
shall be short-term conditioned before the compaction as
8. Sampling and Test Specimen Preparation defined in AASHTO R 30. Prepare the asphalt concrete speci-
8.1 The three-point bending cylinder test may be conducted mens in general accordance with AASHTO R 83 (it takes
on laboratory-prepared test specimens and/or field cores with approximately 16 h for the specimens to be fully cooled down
NMAS less than or equal to 19 mm. to room temperature). Then, core the compacted sample to

4
 D8458 − 22

FIG. 1 General Schematic View of the Three-Point Bending Cylinder Test Setup with a Loaded Specimen: (a) Elevation View; (b) Side
View; and (c) Plan View (continued)

TABLE 1 Test System Minimum Requirements

Measurement Range Accuracy Resolution
Load measurement and control 0 to 5 kN 5N #0.0012 kN
Displacement measurement and control 0 to 5 mm 5 µm 0.0025 mm
Frequency measurement and control 5 to 10 Hz 0.01 Hz Not specified
Temperature measurement and control 5 to 35 °C ±0.5 °C Not specified
Minimum number of data samples for each cycle 40 N/A Not specified

obtain 68 mm diameter samples with an air void level of the PMLC specimens shall not be short-term aged and prepared in
operator’s choice (for example, 7 % 6 0.5 %). Cored samples general accordance with AASHTO R 83. The gyratory speci-
can also be rapidly dried by automatic drying machines (for men will then be cored to obtain 68 mm diameter samples with
example, CoreDry). In this process, a 3PBC sample can be an air void level of the operator’s choice (for example, 7 % 6
tested within an hour of conditioning. Otherwise, traditional 0.5 %). Cored samples can also be rapidly dried by automatic
drying using a fan can be used which needs 24 to 48 h of drying machines (for example, CoreDry). In this process, a
conditioning prior testing. 3PBC sample can be tested within an hour of conditioning.
8.4 Plant-Mixed, Laboratory Compacted (PMLC) Otherwise, traditional drying using a fan can be used which
Specimens—Obtain asphalt concrete samples in accordance needs 24 to 48 h of conditioning prior testing.
with Practice D979/D979M. The 3PBC specimen shall have a 8.5 Field-Cored Specimens—Obtain compacted asphalt
diameter of 68 6 0.5 mm and a minimum length of 150 mm. concrete samples from the roadway in general accordance with

5
 D8458 − 22

FIG. 2 Three-Point Bending Cylinder Test Setup with a Loaded Specimen in the (a) Material Testing System (MTS) and (b) Asphalt
Mixture Performance Tester (AMPT)

TABLE 2 Test System Limits for Data Quality Indicators

Indicator Symbol Equation Limit
Standard error of the applied load se(P) Eq A2.9 #10 %
Average standard error of the measured displacements se(δ) Eq A2.21 #10 %
Uniformity coefficient for the displacement measurements Uδ Eq A2.22 #30 %
Uniformity coefficient for the phase angle measurements Uθ Eq A2.23 #3 degrees

Practice D5361/D5361M. The 3PBC specimen shall be cored 8.6 Measurement of Specimen Dimensions—Measure the
horizontally (perpendicular to the direction of traffic) and shall height and diameter of the specimen to the nearest 0.1 mm at
have a diameter of 68 6 0.5 mm and a minimum length of three different points at 120° apart in accordance with appli-
150 mm. Field coring is only applicable to asphalt concrete cable sections of Test Method D3549/D3549M, determine the
layers thicker than 75 mm. Do not use this method if a average of the measurements for each dimension, and record
homogeneous asphalt concrete layer with thickness greater the averages to the nearest 0.1 mm.
than 75 mm does not exist in the field.

6
 D8458 − 22
8.7 Specimen Preconditioning—Place the specimen in an movement of the side clamps, if applicable. Finally, the top
environmental chamber at a target test temperature 6 1.0 °C bars are placed to prevent any potential deflection of the side
for a minimum of 2 h prior to beginning the test. Exact clamps.4
conditioning time should be determined by using a thermo- 8.11 Set the displacement amplitude based on the desired
couple placed at the center of a dummy sample and recording strain rate by manually adjusting the sensor and the parameters
the time required to reach the target temperature. in the test control software. Select the desired initial strain (for
8.8 Testing Temperatures—Recommended test temperature example, 200 to 800 × 10–6 mm/mm) and loading frequency
is the PG IT defined in Specification D6373, AASHTO M 320, (for example, 5 Hz) and the load cycle intervals at which test
results are to be recorded and computed. Use the following
or AASHTO M 332 and provided in the following equation:
equation to compute the target displacement based on the
PH HT 1PG LT desired strain at the bottom of the 3PBC sample in the
T 1 5 PG IT 5 14 (1)
2 mid-span:
where: πd 3
δz 5 K ~ε !
4L y max
(3)
T1 = first testing temperature (°C),
PGIT = intermediate performance grade temperature (°C), 8.11.1 The parameters of Eq 3 are defined in Section 9, and
PHHT = climatic high-performance grade temperature (°C),
therefore are not repeated here for brevity.
and
PGLT = climatic low-performance grade temperature (°C). 8.12 Within the load cycles to be recorded, include an
NOTE 2—If the data will be analyzed using the simplified viscoelastic interval near the point of five cycles. Average the specimen
continuum damage theory (VECD), test should be repeated at another dynamic modulus at the fifth load cycle; this dynamic modulus
temperature. This is needed to determine if the pseudo stiffness (C) versus is the recommended estimate of the initial beam dynamic
damage parameter (S) relationship is unique, regardless of the temperature modulus.
of testing. Recommended second testing temperature is 10 °C lower than
the T1: 8.13 Select a displacement level (strain level) near 200 ×
10–6 mm/mm initially for the specific material based on trial
T 2 5 T 1 2 10 °C (2)
and error or experience. If a complete fatigue curve is needed
where: for use in the viscoelastic continuum damage (VECD) analysis,
T2 = second testing temperature (°C). adjust the strain up and/or down on additional replicate beams
to evaluate the performance of the material over a range of
8.9 Specimen Load—Open the clamps and place the speci- strain levels.
men into position (Fig. 2). Center the specimen. Once the
specimen is in position, attach the center clamp first. Then 8.14 Upon selection of the appropriate test parameters,
lower the actuator (or raise the actuator, depending on the begin the test. Activate the control and data acquisition system
position of the actuator) so that base plate with two side clamps so that the test results at the selected load cycle intervals are
close to the specimen. Apply a seating load of 40 N (in load monitored and recorded, ensuring that the test system is
control) so that the specimen is seated against the two side operating properly. With low-strain testing, it may be imprac-
tical to reach this desired failure point; in this case, the test can
clamps. Make sure the bottom plate is free to move so that the
be terminated at 100 000 loading cycles or after the specimen
sample can be centered manually in two directions (in the
dynamic modulus reduces to about 20 % of the initial dynamic
direction of diameter and direction of its height). This center-
modulus.
ing process is extremely important. Once centered, tighten the
side C-clamps while in load control mode. Once all clamps are NOTE 3—With some modified materials (for example, polymer-
tightened, tighten the bottom clamp to the testing equipment, modified mixtures, fiber-reinforced asphalt concrete, etc.), lower termina-
tion criteria (for example, when dynamic modulus reduces to less than
again while in load control mode. If there is a gap between the 20 % of the initial dynamic modulus) can be defined.
sample and any of the three clamps, use painter’s masking tape
(for example, 3M Scotch Blue 2080 paper tape) to fill in the
gap. 4
The clamping procedure for the MTS unit is performed in the following order:
8.10 Once the tightening operation is completed, attach the after the three-point bending cylinder setup is attached to the MTS and the specimen
is loaded, apply a contact load of 40 N such that the central clamp aligns on the
two linear variable differential transducers (LVDTs) to the surface at mid-span length of the specimen. Then tighten the screws as described
central clamp. Clamp the LVDTs into position so that it rests on above. For the AMPT unit, initially, the three-point bending cylinder setup is
top of the flat surface of the contact position and check that the attached. The specimen is loaded in the setup and a sliding shaft is used to connect
the central clamp of the setup to the top platen of the AMPT. Once the central clamp
displacement sensor will not overextend its designed length of aligns on the surface of the specimen, tighten the screws as described above then
travel. A third lateral LVDT can be used to monitor the lateral tighten the sliding shaft.

7
 D8458 − 22
9. Calculations method). Therefore, use of a single typical Poisson’s ratio value (for
example, 0.3) in Eq 9 is recommended.
9.1 Perform the following calculations at the operator-
specified load cycle intervals. 9.1.3 Maximum tensile strain:
9.1.1 Maximum tensile stress at the mid-span: ~ σ y! n
εN 5 (10)
M 4P L ?E ?
*
n
~ σ y ! max 5 2 ~ σ y ! min 5 S 5 πdz3 (4)
where:
where: ɛn = maximum tensile strain at each cycle n, and
(σy)max = maximum tensile stress, N/mm2, (σy)n = maximum tensile stress at each cycle n, N/mm2.
(σy)min = minimum tensile stress, N/mm2, 9.1.4 Number of cycles to failure (Nf):
M = maximum bending moment at the center of the 9.1.4.1 Plot the force standard error (Eq A2.9 in Annex A2)
beam (M = PzL/8), N-mm, versus the loading cycles (n) graph, as shown in Fig. 3(a). The
S = section modulus (S = πd3/32), mm3, Nf is the cycle where standard error rapidly increases beyond
Pz = maximum load in each cycle, N, 30 %. An example variation of |E*| versus cycles is provided in
d = diameter, mm, and Fig. 3(b), where Nf is illustrated. When the specimen fails, the
L = span length, mm. shape of the force signal (see Fig. 3(c) for initial cycle) gets
9.1.2 Cylindrical sample dynamic modulus: distorted, as shown in Fig. 3(d). As a result, the standard error
9.1.2.1 Based on Timoshenko beam theory formulations, increases significantly.
maximum deflection (δz) at the center of a cylindrical thick
beam sample has the following relationship: 10. Report
P zL 3 2βP z L ~ 1 1 v ! 10.1 Asphalt Concrete Description—Report the binder type,
δz 5
192EI xx
1
EA
(5) binder content, aggregate gradation, and air void percentage.
10.2 Specimen Dimensions—Report the specimen length,
where:
average specimen height, and average specimen width in
δz = maximum deflection, mm, meters to four significant digits.
A = the cross-sectional area, mm2,
Pz = maximum load in each cycle, N, 10.3 Report the average test temperature to the nearest
Ixx = moment of inertia, mm4, 0.2 °C.
L = span length, mm, 10.4 Report the following test results for each load cycle
E = modulus of elasticity, MPa, interval selected by the operator to three significant figures:
v = Poisson’s ratio, and applied load, beam deflection, tensile stress, tensile strain, and
β = shear coefficient, defined as follows:
dynamic modulus.
6~1 1 v!2 10.5 Report the initial dynamic modulus.
β5 (6)
7112v14v 2
10.6 Report the measured cycles to failure.
9.1.2.2 For viscoelastic materials that are exposed to cyclic
load at a constant frequency, Eq 5 can be rearranged into the 11. Precision and Bias
following form: 11.1 Precision—The within-laboratory repeatability stan-
P z L @ AL 2 1 384 βI xx ~ 1 1 v ! # dard deviation has been determined and is shown in Table 3.
?E ?
*
5
192δ z I xx A
(7) The average coefficient of variation (COV) is 7.4 % and the
maximum is 18.6 %. This data is based on a single lab, eight
9.1.2.3 Further simplification leads to the following equa- test conditions (three replicates for each test condition), and
tion to compute the magnitude of the dynamic modulus at each one asphalt mixture type. The between-laboratory reproduc-
cycle n (for example, |E*|n): ibility of this test method is projected to be completed in
P z~ n ! August 2023. Therefore, this standard should not be used for
?E ?
*
n
5K
δ z~ n !
(8)
acceptance or rejection of a material for purchasing purposes.
L @ AL 2 1 384 βI xx ~ 1 1 v ! # It is to be noted that 8.2 includes the minimum number of
K5 (9) replicates required for this test.
192I xx A
11.2 Bias—No information can be presented on the bias of
where:
the procedure in the three-point bending cylinder (3PBC) test
|E*|n = the (damaged or undamaged) dynamic method for measuring fatigue life because no material having
modulus at each cycle n, MPa, and an accepted reference value is available.
Pz(n) and δz(n) = the peak-to-peak load (N) and deflection
(mm), respectively. 12. Keywords
NOTE 4—A range of typical Poisson’s ratio values based on NMAS and
PG are listed in Annex A1 in Table A1.1. It is noted that the choice of 12.1 asphalt concrete dynamic modulus; asphalt concrete
Poisson’s ratio shifts the |E*| versus Nf graph up or down, but it does not fatigue; asphalt concrete tensile testing; fatigue life; flexural
affect the magnitude of Nf (which is the main outcome of this test bending; shear bending; Timoshenko beam theory

8
 D8458 − 22

FIG. 3 Example Graphs of (a) Force Standard Error (se(P)) versus Cycles (n); (b) Dynamic Modulus (|E*|) versus Cycles (n); (c) Force
(P) versus Time at the Initial Cycle; and (d) Force (P) versus Time at the Final Cycle

TABLE 3 Single-Laboratory Precision Data

NOTE 1—Descriptions of the abbreviated words in the table headings are as follows: Rep = replicate number, AV = air voids, T = temperature, f =
frequency, Nf = number of cycles to failure, Avg. = average, Std. = standard deviation, d2s = difference two-sigma limit, COV = coefficient of variation.
Placement of
Strain Clamp Nf Nf Avg. Std. d2s
Test No. AV (%) T (°C) f (Hz) Central Clamp COV (%)
Level (µϵ) Tightness (Rep 1) (Rep 2) Nf Nf Nf
(Eccentricity)
1 6% 400 Full 20 5 zero eccentricity 6687 7260 6974 405 1145 5.8 %
2 4% 400 Partial 10 10 zero eccentricity 8077 9223 8650 810 2292 9.4 %
3 4% 400 Partial 20 5 6 mm eccentricity 4917 3774 4346 808 2286 18.6 %
4 6% 200 Partial 20 10 zero eccentricity 489 426 458 44 125 9.7 %
5 4% 200 Full 20 10 6 mm eccentricity 5009 4974 4991 25 70 0.5 %
6 6% 400 Full 10 10 6 mm eccentricity 29111 29459 29285 246 696 0.8 %
7 6% 200 Partial 10 5 6 mm eccentricity 12299 14930 13615 1860 5262 13.7 %
8 4% 200 Full 10 5 zero eccentricity 18915 19028 18972 80 226 0.4 %

Average = 7.4 %

ANNEXES

(Mandatory Information)

A1. TYPICAL POISSON’S RATIO VALUES FOR ASPHALT MIXTURES

A1.1 Table A1.1 shows the typical Poisson’s ratio values TABLE A1.1 Recommended Poisson’s Ratio Values for the 3PBC
Test Data Analysis
that can be used to analyze 3PBC test data, for different NMAS
and PG combinations. Performance Nominal Maximum Aggregate Size (mm)
Grade (PG) 9.5 12.5 19.0 25.0
PG 58-XX 0.28–0.35 0.28–0.35 0.25–0.32 0.25–0.32
PG 64-XX 0.25–0.30 0.25–0.30 0.25–0.30 0.25–0.30
PG 70-XX 0.20–0.28 0.20–0.28 0.20–0.28 0.20–0.28
PG 76-XX 0.15–0.25 0.15–0.25 0.15–0.25 0.15–0.25
PG 82-XX 0.15–0.25 0.15–0.25 0.15–0.25 0.15–0.25

9
 D8458 − 22

A2. CALCULATIONS FOR DATA QUALITY STATISTICS

A2.1 The general approach used herein is based on the where:

least-squares fit of a sinusoid, as described by Chapra and AP0 = load offset coefficient, N (lb),
Canale in Numerical Methods for Engineers (McGraw-Hill, AP1 = load in-phase magnitude coefficient, N (lb),
1985, pp. 404–407). This approach can be performed on a BP1 = load out-of-phase magnitude coefficient, N (lb),
spreadsheet or programmed into an algorithm of the user’s P'i = centered load at point i in the data array, N (lb),
choice. ω0 = frequency of applied load, rad/s, and
ti = time at point i in the data array, s.
A2.2 The load and displacement data produced from each
3PBC test at an angular frequency ω0 (ω0 = 2πf, where f = A2.3.4 From the load coefficients, compute the load mag-
frequency in Hz) are stored in the form of several arrays, one nitude and the load phase angle.
for time [ti], one for load [Pi], and one for each of the j = 1, 2
displacement transducers used [δj]. The number of i = 1, 2,
? P ? 5 =A
* 2
P1 1B 2P1 (A2.6)

3…n points in each array will be equal, and will depend upon
the number of data points collected per loading cycle and on
?θ ?P 5 arctan 2SB P1
A P1 D (A2.7)

the total number of cycles for which data has been collected. where:
Each time the testing system records the data, it is recom- |P*| = load magnitude, N (lb),
mended that at least 40 points per cycle be used. θP = load phase angle, degrees,
AP1 = load in-phase magnitude coefficient, N (lb), and
A2.3 Analyze Load Data—The first step is to analyze the BP1 = load out-of-phase magnitude coefficient, N (lb).
data in the load array. The data analysis is performed on
centered load data which is computed from the raw load data A2.3.5 Compute an array of predicted centered loads and
by subtracting the average load. the standard error of the applied load.
A2.3.1 Determine the average load as: P̂' i 5 A P0 1A P1 cos~ ω 0 t ! 1B P1 sin~ ω 0 t ! (A2.8)
n
n
(P
!
2
i
( ~ P̂' 2 P' !
P̄ 5
i51

n
(A2.1) se ~ P ! 5
i51
i

n24
i

S? ?D
100%
P*
(A2.9)

where: where:
P̄ = average load, N (lb), P̂'i = predicted centered load at point i, N (lb),
Pi = raw load point i in the data array, N (lb), and se(P) = standard error for the applied load, %,
n = number of points in the data array. AP0 = load offset coefficient, N (lb),
A2.3.2 Then compute the centered loads by subtracting the AP1 = load in-phase magnitude coefficient, N (lb),
average load from each of the load measurements: BP1 = load out-of-phase magnitude coefficient, N (lb),
P'i = centered load at point i in the data array, N (lb),
p' i 5 P i 2 P̄ (A2.2) ω0 = frequency of applied load, rad/s,
ti = time at point i in the data array, s,
where: |P*| = amplitude of the cyclic load data (half of peak-to-
P'i = centered load at point i in the data array, N (lb), peak), N (lb), and
P̄ = average load, N (lb), and n = number of points in data array.
Pi = raw load point i in the data array, N (lb).
A2.4 Analyze Displacement Data—The second step is to
A2.3.3 From the centered load data, compute three load
perform a similar analysis on the data from each of the
coefficients: offset, in-phase magnitude, and out-of-phase mag-
displacement transducers. However, in this case the data are
nitude.
corrected for drift caused by permanent deformation during the
n
test and centered data based on the average displacement for
( P'
i51
i
the transducer.
A P0 5 (A2.3)
n
A2.4.1 To estimate the drift in the displacement data, search
2 n
A P1 5 ( P' cos~ ω 0 t i ! (A2.4) each displacement transducer array and determine local maxi-
n i51 i
mum and minimum values and the time when they occur for
2 n each loading cycle. Then determine the slope of the local
B P1 5 ( P' sin~ ω 0 t i !
n i51 i
(A2.5)
maximum and minimum values with respect to time using

10
 D8458 − 22
linear regression. The average of these two slopes is the rate of
drift Dj for displacement transducer j. ? θ ? 5 arctan
δj S 2
B δj1
A δj1 D (A2.16)

A2.4.2 Determine the average displacement for each dis- where:

placement transducer as: |δj*| = displacement magnitude for displacement transducer
n
j,
(δ
i51
ji θδj = phase angle for displacement transducer j, degrees,
δ̄ j 5 (A2.10) Aδj1 = in-phase magnitude coefficient for displacement
n
transducer j, and
where: Bδj1 = out-of-phase magnitude coefficient for displacement
δ̄j = average displacement for transducer j, transducer j.
δji = raw displacement for transducer j at point i in data
array, and A2.4.6 For each displacement transducer, compute an array
n = number of points in the data array. of predicted corrected and centered displacements and the
standard error of the displacement data.
A2.4.3 The displacement data for each transducer are then
corrected and centered by subtracting from the measured δ̂' ji 5 A δj0 1A δj1 cos~ ω 0 t ! 1B δj1 sin~ ω 0 t ! (A2.17)
displacements the rate of drift times the loading time and also n
subtracting the average displacement for that transducer:
!
2
( ~ δ̂' 2 δ' ji !
δ' ji 5 δ ji 2 D j t i 2 δ̄ j (A2.11) se ~ δ j ! 5
i51
ji

n24 S? ?D
100%
δ *j
(A2.18)

where: where:
δ'ji = corrected and centered displacement for transducer j at δ̂'ji = predicted corrected and centered displacement for
point i in data array, displacement transducer j at point i,
δji = raw displacement for transducer j at point i in data se(δj) = standard error for displacement transducer j
array, response, %,
ti = time for point i in data array, s, Aδj0 = offset coefficient for displacement transducer j,
δ̄j = average displacement for transducer j, and Aδj1 = in-phase magnitude coefficient for displacement
Dj = rate of drift for transducer j. transducer j,
A2.4.4 From the corrected and centered displacement data Bδj1 = out-of-phase magnitude coefficient for displacement
for each displacement transducer, compute three displacement transducer j,
coefficients: offset, in-phase magnitude, and out-of-phase mag- δ'ji = corrected and centered displacement for transducer j
nitude. at point i in data array,
n
ti = time for point i in data array, s,
ω0 = frequency of applied load, rad/s,
( δ'
i51
ji
|δj*| = amplitude of the cyclic displacement transducer j
A δj0 5 (A2.12)
n data (half of peak-to-peak), and
2 n n = number of points in data array.
A δj1 5 ( δ' cos~ ω 0 t i !
n i51 ji
(A2.13)
A2.4.7 Then, the average phase angle, displacement
2 n magnitude, and standard error are calculated for all m displace-
B δj1 5 (
δ' sin~ ω 0 t i !
n i51 ji
(A2.14)
ment transducers, along with two uniformity coefficients rep-
where: resenting the variation among the displacement transducers:
m
Aδj0 = offset coefficient for displacement transducer j,
Aδj1 = in-phase magnitude coefficient for displacement (θ
j51
δj
θ¯δ 5 (A2.19)
transducer j, m
Bδj1 = out-of-phase magnitude coefficient for displacement m

transducer j, ( ?δ ? *
j
δ'ji = corrected and centered displacement for transducer j
at point i in data array,
? ?
δ¯* 5
j51

m
(A2.20)

m
ti = time for point i in data array, s, and
ω0 = frequency of applied load, rad/s. ( se ~ δ !
j51
j
se ~ δ ! 5 (A2.21)
A2.4.5 From the displacement coefficients, compute the m
displacement magnitude and the displacement phase angle for m

( ~ ? δ ? 2 ? δ¯ ? !
!
2
* *
each transducer.

? δ ? 5 =A
*
j
2
δj1 1B 2
δj1 (A2.15)
Uδ 5
j51
j

m21 S? ?D
100%
δ¯*
(A2.22)

11
 D8458 − 22
m se(δ) = average standard error for all displacement

Uθ 5 ! ( ~θ
j51
δj

m21
2 θ δ! 2
(A2.23) Uδ
transducers, %,
= uniformity coefficient for displacement transducers,
%,
where: Uθ = uniformity coefficient for phase, degrees, and
θ̄δ = average phase angle for all displacement transducers, m = number of displacement transducers.
degrees,
|δ*| = average amplitude of the cyclic displacement (half of
peak-to-peak),

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