ASTM D4015-21

Standard Test Methods for Modulus and Damping of Soils by Fixed-Base Resonant Column Devices

SIGNIFICANCE AND USE
5.1 The equivalent elastic shear modulus and damping capacity of a given soil, as measured by the resonant column technique herein described, depend upon the strain amplitude of vibration, the state of effective stress, and the void ratio of the soil, temperature, time, etc. Since the application and control of the static axial and lateral stresses and the void ratio are not prescribed in these methods, the applicability of the results to field conditions will depend on the degree to which the application and control of the static axial and lateral stresses and the void ratio, as well as other parameters such as soil structure, duplicate field conditions. The techniques used to simulate field conditions depend on many factors and it is up to the engineer to decide on which techniques apply to a given situation and soil type. The results of these tests are useful for calculations involving soil-structure interaction and seismic response of soil deposits.
Note 1: The quality of the results produced by this standard is dependent on the competence of the personnel performing it, and the suitability of the equipment and facilities. Agencies that meet the criteria of Practice D3740 are generally considered capable of competent and objective testing/sampling/inspection/etc. Users of this standard are cautioned that compliance with Practice D3740 does not in itself assure reliable results. Reliable results depend on many factors; Practice D3740 provides a means of evaluating some of those factors.
SCOPE
1.1 These test methods cover the determination of shear modulus and shear damping as a function of shear strain amplitude for solid cylindrical specimens of soil in intact and reconstituted conditions by torsional vibration using resonant column devices. The vibration of the specimen may be superposed on a controlled static state of stress in the specimen. The vibration apparatus and specimen may be enclosed in a triaxial chamber and subjected to an all-around pressure and axial load. In addition, the specimen may be subjected to other controlled conditions (for example, pore-water pressure, degree of saturation, temperature). These test methods of modulus and damping determination are considered nondestructive when the shear strain amplitudes of vibration are less than 10–2 % (10–4 in./in.), and many measurements may be made on the same specimen and with various states of static stress.  
1.2 Two device configurations are covered by these test methods: Device Type 1 where a known torque is applied to the top of the specimen and the resulting rotational motion is measured at the top of the specimen, and Device Type 2 where an uncalibrated torque is applied to the top of the specimen and the torque transmitted through the specimen is measured by a torque transducer at the base of the specimen. For both types of devices, the torque is applied to the active end (usually top) of the specimen and the rotational motion also is measured at the active end of the specimen.  
1.3 These test methods are limited to the determination of the shear modulus and shear damping, the necessary vibration, and specimen preparation procedures related to the vibration, etc., and do not cover the application, measurement, or control of the axial and lateral static normal stresses. The latter procedures may be covered by, but are not limited to, Test Method D2850, D4767, or D7181.  
1.4 Significant Digits—All recorded and calculated values shall conform to the guide for significant digits and rounding established in Practice D6026.  
1.4.1 The procedures used to specify how data are collected/recorded and calculated in this standard are regarded as the industry standard. In addition, they are representative of the significant digits that should generally be retained. The procedures used do not consider material variation, purpose for obtaining the data, special purpose studies, or any considerations for the user’s objec...

General Information

• Status: Published
• Publication Date: 30-Nov-2021
• ICS: 93.020 - Earthworks. Excavations. Foundation construction. Underground works
• Technical Committee: D18 - Soil and Rock
• Drafting Committee: D18.09 - Cyclic and Dynamic Properties of Soils

Relations

• Refers: ASTM D4753-24 - Standard Guide for Evaluating, Selecting, and Specifying Balances and Standard Masses for Use in Soil, Rock, and Construction Materials Testing
• Effective Date: 01-Feb-2024
• Refers: ASTM D3740-23 - Standard Practice for Minimum Requirements for Agencies Engaged in Testing and/or Inspection of Soil and Rock as Used in Engineering Design and Construction
• Effective Date: 01-Nov-2023
• Refers: ASTM D4767-11(2020) - Standard Test Method for Consolidated Undrained Triaxial Compression Test for Cohesive Soils
• Effective Date: 01-Apr-2020
• Refers: ASTM D7181-20 - Standard Test Method for Consolidated Drained Triaxial Compression Test for Soils
• Effective Date: 01-Jan-2020
• Refers: ASTM D3740-19 - Standard Practice for Minimum Requirements for Agencies Engaged in Testing and/or Inspection of Soil and Rock as Used in Engineering Design and Construction
• Effective Date: 01-Oct-2019
• Refers: ASTM D2216-19 - Standard Test Methods for Laboratory Determination of Water (Moisture) Content of Soil and Rock by Mass
• Effective Date: 01-Mar-2019
• Refers: ASTM D4753-15 - Standard Guide for Evaluating, Selecting, and Specifying Balances and Standard Masses for Use in Soil, Rock, and Construction Materials Testing
• Effective Date: 01-May-2015
• Refers: ASTM D653-14 - Standard Terminology Relating to Soil, Rock, and Contained Fluids
• Effective Date: 01-Aug-2014
• Refers: ASTM D2166/D2166M-13 - Standard Test Method for Unconfined Compressive Strength of Cohesive Soil
• Effective Date: 15-May-2013
• Refers: ASTM D3740-12a - Standard Practice for Minimum Requirements for Agencies Engaged in Testing and/or Inspection of Soil and Rock as Used in Engineering Design and Construction
• Effective Date: 01-May-2012
• Refers: ASTM D3740-12 - Standard Practice for Minimum Requirements for Agencies Engaged in Testing and/or Inspection of Soil and Rock as Used in Engineering Design and Construction
• Effective Date: 01-Mar-2012
• Refers: ASTM D653-11 - Standard Terminology Relating to Soil, Rock, and Contained Fluids
• Effective Date: 01-Sep-2011
• Refers: ASTM D3740-11 - Standard Practice for Minimum Requirements for Agencies Engaged in Testing and/or Inspection of Soil and Rock as Used in Engineering Design and Construction
• Effective Date: 01-Sep-2011
• Refers: ASTM D7181-11 - Method for Consolidated Drained Triaxial Compression Test for Soils
• Effective Date: 01-Jul-2011
• Refers: ASTM D2216-10 - Standard Test Methods for Laboratory Determination of Water (Moisture) Content of Soil and Rock by Mass
• Effective Date: 01-Jul-2010

Overview

ASTM D4015-21: Standard Test Methods for Modulus and Damping of Soils by Fixed-Base Resonant Column Devices provides an established procedure for determining the shear modulus and damping characteristics of soils. These properties are key to understanding the dynamic and cyclic behavior of soil deposits under various loading conditions, such as earthquakes or machine vibrations. The standard outlines methods for using fixed-base resonant column devices to apply torsional vibrations to cylindrical soil specimens, measuring soil response, and determining stiffness and energy dissipation characteristics.

The test methods are widely recognized for their non-destructive approach, allowing repeated measurements of soil modulus and damping across a range of strain amplitudes and static stress states without significantly altering specimen properties.

Key Topics

• Shear Modulus and Damping: The standard specifies how to measure soil's elastic shear modulus and damping capacity-critical parameters for evaluating soil response to dynamic loading.

• Device Configurations: Two device types are addressed:

  • Device Type 1: Applies a known torque and measures top-end rotation.
  • Device Type 2: Applies uncalibrated torque and measures transmitted torque at the base.

• Test Conditions: Soil samples may be tested under varying levels of confining pressure, axial load, temperature, saturation, and pore water pressure for relevance to field conditions.

• Specimen Preparation: Focuses on maintaining sample integrity and proper coupling between soil and device platens, with specific requirements for size, shape, and end conditions.

• Reporting and Data Quality: Emphasizes traceability, calibration, and correct use of significant digits in test results, as well as the importance of competent personnel and reliable instrumentation.

Applications

The procedures outlined in ASTM D4015-21 are widely used in geotechnical engineering and seismic site characterization. Key applications include:

• Soil-Structure Interaction Analysis: Accurate modulus and damping values are essential for evaluating how soil will interact with foundations and structures during seismic or vibrational events.

• Seismic Response Evaluation: Supports modeling of ground motion amplification and wave propagation through soil layers.

• Design of Foundations and Earthworks: Designers use dynamic soil properties to optimize the safety and efficiency of building foundations, retaining walls, and embankments.

• Research and Development: Researchers employ resonant column methods to investigate soil behavior under cyclic loading, improve ground improvement techniques, and calibrate advanced numerical models.

• Quality Control in Laboratory Testing: The standard ensures consistency and repeatability in measurements of dynamic soil properties, supporting robust data for engineering projects.

Related Standards

ASTM D4015-21 references and complements several other ASTM standards used in soil testing and geotechnical site investigation:

• ASTM D653: Terminology Relating to Soil, Rock, and Contained Fluids
• ASTM D2166/D2166M: Test Method for Unconfined Compressive Strength of Cohesive Soil
• ASTM D2850: Test Method for Unconsolidated-Undrained Triaxial Compression Test on Cohesive Soils
• ASTM D3740: Practice for Agencies Engaged in Testing/Inspection of Soil and Rock
• ASTM D4767: Consolidated Undrained Triaxial Compression Test for Soils
• ASTM D6026: Practice for Using Significant Digits and Data Records in Geotechnical Data

Practical Value

ASTM D4015-21 offers a robust, standardized approach for laboratories and engineering professionals to quantify and compare dynamic properties of soils. By enabling accurate, repeatable measurements of soil modulus and damping, it provides critical input for seismic design, soil-structure interaction modeling, and advanced geotechnical analysis, supporting safer and more resilient infrastructure development. Adherence to this standard also ensures consistency and comparability of results across laboratories and projects worldwide, facilitating best practices in geotechnical site characterization and soil dynamics.

Standards Content (Sample)

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: D4015 − 21
Standard Test Methods for
Modulus and Damping of Soils by Fixed-Base Resonant
Column Devices
This standard is issued under the fixed designation D4015; 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* procedures may be covered by, but are not limited to, Test
Method D2850, D4767,or D7181.
1.1 These test methods cover the determination of shear
modulus and shear damping as a function of shear strain 1.4 Significant Digits—All recorded and calculated values
amplitude for solid cylindrical specimens of soil in intact and shall conform to the guide for significant digits and rounding
reconstituted conditions by torsional vibration using resonant established in Practice D6026.
column devices. The vibration of the specimen may be 1.4.1 Theproceduresusedtospecifyhowdataarecollected/
superposed on a controlled static state of stress in the speci- recorded and calculated in this standard are regarded as the
men.Thevibrationapparatusandspecimenmaybeenclosedin industry standard. In addition, they are representative of the
a triaxial chamber and subjected to an all-around pressure and significant digits that should generally be retained. The proce-
axial load. In addition, the specimen may be subjected to other dures used do not consider material variation, purpose for
controlled conditions (for example, pore-water pressure, de- obtaining the data, special purpose studies, or any consider-
gree of saturation, temperature). These test methods of modu- ations for the user’s objectives; and it is common practice to
lus and damping determination are considered nondestructive increase or reduce significant digits of reported data to be
when the shear strain amplitudes of vibration are less than commensuratewiththeseconsiderations.Itisbeyondthescope
–2 –4
10 %(10 in.⁄in.),andmanymeasurementsmaybemadeon of this standard to consider significant digits used in analysis
the same specimen and with various states of static stress. methods for engineering design.
1.4.2 Measurements made to more significant digits or
1.2 Two device configurations are covered by these test
better sensitivity than specified in this standard shall not be
methods:DeviceType1whereaknowntorqueisappliedtothe
regarded a nonconformance with this standard.
top of the specimen and the resulting rotational motion is
measured at the top of the specimen, and Device Type 2 where 1.5 Units—The values stated in SI units are to be regarded
anuncalibratedtorqueisappliedtothetopofthespecimenand as standard. The values given in parentheses are mathematical
the torque transmitted through the specimen is measured by a conversions to inch-pound units, which are provided for
torquetransduceratthebaseofthespecimen.Forbothtypesof information only and are not considered standard. Reporting of
devices, the torque is applied to the active end (usually top) of test results in units other than SI shall not be regarded as
the specimen and the rotational motion also is measured at the nonconformance with these test methods.
active end of the specimen. 1.5.1 The converted inch-pound units use the gravitational
systemofunits.Inthissystem,thepound(lbf)representsaunit
1.3 These test methods are limited to the determination of
of force (weight), while the unit for mass is slugs. The
the shear modulus and shear damping, the necessary vibration,
converted slug unit is not given, unless dynamic (F = ma)
and specimen preparation procedures related to the vibration,
calculations are involved.
etc., and do not cover the application, measurement, or control
1.5.2 It is common practice in the engineering/construction
of the axial and lateral static normal stresses. The latter
profession to concurrently use pounds to represent both a unit
of mass (lbm) and of force (lbf). This implicitly combines two
separate systems of units; that is, the absolute system and the
These test methods are under the jurisdiction ofASTM Committee D18 on Soil
and Rock and are the direct responsibility of Subcommittee D18.09 on Cyclic and
gravitational system. It is scientifically undesirable to combine
Dynamic Properties of Soils.
the use of two separate sets of inch-pound units within a single
Current edition approved Dec. 1, 2021. Published December 2021. Originally
ɛ1 standard. As stated, this standard includes the gravitational
approved in 1981. Last previous edition approved in 2015 as D4015 –15 . DOI:
10.1520/D4015-21. system of inch-pound units and does not use/present the slug
*A Summary of Changes section appears at the end of this standard
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
D4015 − 21
unit for mass. However, the use of balances or scales recording D3740 Practice for Minimum Requirements for Agencies
pounds of mass (lbm) or recording density in lbm/ft shall not Engaged in Testing and/or Inspection of Soil and Rock as
be regarded as nonconformance with this standard. Used in Engineering Design and Construction
D4753 Guide for Evaluating, Selecting, and Specifying Bal-
1.6 This standard does not purport to address all of the
ances and Standard Masses for Use in Soil, Rock, and
safety concerns, if any, associated with its use. It is the
Construction Materials Testing
responsibility of the user of this standard to establish appro-
D4767 Test Method for Consolidated Undrained Triaxial
priate safety, health, and environmental practices and deter-
Compression Test for Cohesive Soils
mine the applicability of regulatory limitations prior to use.
D6026 Practice for Using Significant Digits and Data Re-
1.7 This international standard was developed in accor-
cords in Geotechnical Data
dance with internationally recognized principles on standard-
D7181 Test Method for Consolidated DrainedTriaxial Com-
ization established in the Decision on Principles for the
pression Test for Soils
Development of International Standards, Guides and Recom-
mendations issued by the World Trade Organization Technical
3. Terminology
Barriers to Trade (TBT) Committee.
3.1 Definitions—For definitions of common technical terms
2. Referenced Documents
used in this standard, refer to Terminology D653.
2.1 ASTM Standards:
3.2 Definitions of Terms Specific to This Standard:
D653 Terminology Relating to Soil, Rock, and Contained
3.2.1 damping capacity D [unitless, typically expressed in
Fluids
%], n—in resonant column systems,isrelatedtothecomponent
D2166/D2166M Test Method for Unconfined Compressive
of the dynamic shear modulus that lags the applied shear stress
Strength of Cohesive Soil
by 90° degrees.
D2216 Test Methods for Laboratory Determination of Water
3.2.2 Device Type 1, DT1, n—in resonant column systems,a
(Moisture) Content of Soil and Rock by Mass
resonant column system as shown in Fig. 1 where the passive
D2850 Test Method for Unconsolidated-Undrained Triaxial
end platen is directly connected to the Fixed Base (no torque
Compression Test on Cohesive Soils
transducer), a calibrated vibratory torque is applied to the
active end, and rotation is measured at the active end.
3.2.2.1 Discussion—The vibration excitation device may
incorporate springs and dashpots connected to the active-end
For referenced ASTM standards, visit the ASTM website, www.astm.org, or platen, where the spring constants and viscous damping coef-
contact ASTM Customer Service at [e-mail]. For Annual Book of ASTM
ficients must be known. The rotational inertia of the active-end
Standards volume information, refer to the standard’s Document Summary page on
the ASTM website.
For Device Type 1, no torque transducer is needed and the Passive End Platen is connected to the Fixed Base.
FIG. 1 Resonant-Column Schematic for Both Device Types 1 and 2
D4015 − 21
platen and portions of the vibration excitation device moving frequencies other than resonant frequencies. Given the
with it must be known. geometry, mass and system parameters, the equivalent elastic
shear modulus and damping capacity can be determined at a
3.2.3 Device Type 2, DT2, n—in resonant column systems,a
measured level of excitation vibration. The amplitude of
resonant column system as shown in Fig. 1 where the passive
vibration (which is related to shear strain) is typically varied to
end platen is connected to a torque transducer, an uncalibrated
measure the variation of modulus and damping as a function of
torque is applied to the active end, torque is measured by the
shear strain. The test is usually conducted at levels of shear
torque transducer at the passive end, and rotation is measured
strain between 0.00001 % and 0.2 %. (The upper limit of shear
at the active end.
strain is dependent on the specimen stiffness and the maximum
3.2.3.1 Discussion—The vibration excitation device may
torque capability of the excitation system.) For specimens
incorporate springs and dashpots connected to the active-end
where the maximum shear strain measured is of the order of
platen, but the spring constants and viscous damping coeffi-
0.01 %, the test is often conducted at several different sets of
cients are not needed. The rotational inertia of the active-end
static axial and lateral stress conditions to measure the varia-
platen and portions of the vibration excitation device moving
tion of moduli and damping with static stress states. The test
with it also are not needed.
results are dependent on sample quality/specimen disturbance
3.2.4 dimensionless specimen stiffness, DSS* [unitless],
which are beyond the scope of this standard.
n—in resonant column systems, is a complex number used to
characterize the real and imaginary components of the speci-
5. Significance and Use
men stiffness.
5.1 The equivalent elastic shear modulus and damping
-2
3.2.5 dynamic shear modulus, G* [FL ], n—in resonant
capacity of a given soil, as measured by the resonant column
column systems, is the ratio of shear stress to shear strain under
technique herein described, depend upon the strain amplitude
vibratory conditions (also known as complex shear modulus).
of vibration, the state of effective stress, and the void ratio of
-2
3.2.6 equivalent elastic shear modulus G [FL ], n—in
the soil, temperature, time, etc. Since the application and
resonant column systems, is the component of the dynamic
control of the static axial and lateral stresses and the void ratio
shear modulus that is in-phase with the applied shear stress.
are not prescribed in these methods, the applicability of the
results to field conditions will depend on the degree to which
3.2.7 resonant-column system, n—a system as shown in Fig.
the application and control of the static axial and lateral
1 consisting of a cylindrical specimen or column of soil
stresses and the void ratio, as well as other parameters such as
enclosed with a flexible membrane that has platens attached to
soil structure, duplicate field conditions. The techniques used
each end and where a sinusoidal vibration excitation device is
tosimulatefieldconditionsdependonmanyfactorsanditisup
attached to the active-end platen and where the other end is the
to the engineer to decide on which techniques apply to a given
passive-end platen that is rigidly fixed.
situation and soil type. The results of these tests are useful for
3.2.8 specimen shear strain γ, [unitless, frequently ex-
calculations involving soil-structure interaction and seismic
pressed as %],n—in resonant column systems, is the average
response of soil deposits.
shearstraininthespecimenwheretheshearstrainineachcross
section varies from zero along the axis of rotation to a
NOTE 1—The quality of the results produced by this standard is
dependent on the competence of the personnel performing it, and the
maximum at the perimeter of the specimen.
suitability of the equipment and facilities. Agencies that meet the criteria
3.2.8.1 Discussion—The radius for calculating average
of Practice D3740 are generally considered capable of competent and
shear strains vary depending on soil type, strain level, confin-
objective testing/sampling/inspection/etc. Users of this standard are cau-
ing stress, etc. The default value of the radius for calculating
tioned that compliance with Practice D3740 does not in itself assure
average strain is 0.4*diameter but values in the range of 0.33 reliable results. Reliable results depend on many factors; Practice D3740
provides a means of evaluating some of those factors.
to 0.40*diameter may be used if the value is documented in the
report.
6. Apparatus
-1
3.2.9 system resonant frequency f [s ], n—in resonant col-
r
6.1 General—The complete test apparatus is shown sche-
umn systems, for Device Type 1 is the lowest frequency at
matically in Fig. 1 and includes the platens for holding the
which the rotational velocity at the active end is in phase with
specimen in the pressure cell, the vibration excitation device
the sinusoidal excitation torque and for Device Type 2, is the
(torque motor), transducers for measuring the response, the
lowest frequency at which the rotational motion at the active
control and readout instrumentation, and auxiliary equipment
end is a maximum.
for specimen preparation. The theory for the resonant column
4. Summary of Test Method is provided in Annex A1. The entire apparatus is generally
enclosed within a pressure chamber (commonly referred to as
4.1 The resonant column device is shown schematically in
a triaxial cell). For some apparatus that can apply an axial load
Fig. 1. In the resonant column test, a cylindrical soil specimen,
to the specimen, the pressure chamber lid may be fitted with a
usually enclosed with a thin membrane, is subjected to an
piston passing through the top.
imposed static axial and lateral stress condition. Torsional
sinusoidal vibrations are applied at the top of the soil specimen 6.2 Specimen Platens—Both the active-end and passive-end
and the rotational response is measured. The frequency of platens shall be constructed of noncorrosive material having a
excitation is varied until the system resonant frequency is modulus at least ten times the modulus of the material to be
achieved as described in 3.2.9. The devices may be operated at tested. Each platen shall have a circular cross section and a
D4015 − 21
testing frequencies vary over a wide range. Use of several calibration rods
plane surface of contact with the specimen, except that the
with differing torsional stiffness may be needed.
planesurfaceofcontactshallberoughenedtoprovideformore
efficient coupling with the ends of the specimen. Roughening
6.4 Passive End Torque Transducer—Thistorquetransducer
and flow of fluids into or from the specimen shall be accom-
for Device Type 2 must be waterproof and insensitive to
plished by rigidly fastening porous disks to the platens. The
ambient pressure and temperature changes for the expected
diameter of platens shall be equal to or greater than the
values. It may be a transducer that also measures axial force.
diameter of the specimen. The construction of the platens shall
The torque transducer must have a torque capacity of at least
be such that their stiffness is at least ten times the stiffness of
twice the maximum torque capability of the vibration excita-
the specimen.
tion device, a linearity of 60.5 % of full-scale output, hyster-
6.2.1 The active-end platen may have a portion of the
esis less than 60.1 % of full-scale output, and repeatability
excitation device, transducers, springs, and dashpots connected
better than 60.5 % of full-scale output. If the transducer is
to it. The transducers and moving portions of the excitation
used to measure axial force, the specifications must be similar
device must be connected to the platen in such a fashion that
tothosefortorque.Thetransducermustberigidlyconnectedto
they are to be considered part of the platen, be counterbalanced
the chamber base and the sensing head of the torque transducer
to maintain rotational symmetry, and have the same motion as
shall be rigidly connected to the passive end platen.
the platen for the full range of frequencies to be encountered
6.5 Sine Wave Generator—The sine wave generator is an
when testing soils.
electronic instrument capable of producing a sinusoidal current
6.2.2 The theoretical model used for the resonant-column
with a means of adjusting the frequency over the entire range
system represents the active-end platen, with all attachments,
of operating frequencies anticipated. This instrument shall
as a rigid mass that is attached to the specimen; this mass may
provide sufficient power to produce the desired vibration
alsohavemasslessspringsanddashpotsattachedtoitasshown
amplitude, or its output may be electronically amplified to
in Fig. 1. If springs are used, the excitation device and
provide sufficient power.
active-end platen (without the specimen in place) form a
single-degree-of-freedom system having an undamped natural
6.6 Vibration-Measuring Devices and Readout
frequency, f .
a Instruments—Thesedevicesandinstrumentsshallbecalibrated
6.2.3 The passive-end platen must be rigidly fixed. It may
withanaccuracyof5 %andmustbetraceabletoagovernment
be assumed to be rigidly fixed when the inertia of it and the
standards agency. The vibration-measuring devices shall be
mass(es) attached to it are at least 500 times the inertia of the
acceleration, velocity, or displacement transducers that can be
active-end platen (1) .
attached to and become a part of the active-end platen. The
6.2.4 For Device Type 2, a torque transducer is placed
transducer(s) shall be mounted to produce a calibrated electri-
between the passive end platen and the rigidly fixed base. The
cal output that is proportional to the rotational acceleration,
torque transducer even though relatively stiff in torsion (see
velocity, or displacement. If the rotation is measured using
6.4), must allow for some small rotation of the passive end
linear motion measurement transducers acting at a radius from
platen in order to register the transmitted torque. The inertia of
the axis of rotation, two transducers must be used with both at
thepassiveendplatensystem, J mustincludetheinertiaofthe
the same radius from the axis of rotation but connected
p
sensing head of the torque transducer which is rigidly fastened
diagonally across the axis of rotation and they must be wired
to it. With no specimen in place, the passive end platen system
such that rotational motion is additive and lateral translation is
inertia, J , along with the stiffness, k , and damping coefficient,
p p subtracted (Note 3). The readout instruments must have a
c , of the torque transducer constitute a single-degree-of-
frequency resolution of at least 0.1 Hz. It also is necessary to
p
freedom system which are accounted for in Eq A1.3.
have an electronic device for establishing the phase difference
between the applied and/or measured torque and resulting
6.3 Vibration Excitation Device (torque motor)—This shall
rotational motion.
be a device capable of applying a sinusoidal torsional vibration
totheactive-endplatentowhichthemovingpartsofthedevice
NOTE 3—The use of two linear motion transducers used to measure
are rigidly coupled. The frequency of excitation shall be
rotation will minimize resonances in the bending modes from being
continuously variable and have a range that typically includes
confused with those for the torsional modes.
10 Hz to 1 kHz. For Device Type 1 where the torque is
6.6.1 For Device Type 1, two types of tests are commonly
measured at the active end, the excitation device shall have a
used: 1) steady-state vibration where applied frequency is
means of measuring the torque applied to the excitation device
varied. (Adual channel digital oscilloscope, readout device, or
that has at least 5 % accuracy of full-scale output. If an
spectrum analyzer may be used.) and 2) transient vibration
electromagnetic excitation device is used, the voltage drop
starting from a steady-state vibration at the resonant frequency
across a fixed, temperature-and-frequency-stable power resis-
and excitation is then cut off and the decay of vibratory motion
tor in series with the excitation device is proportional to
is measured with time (a digital x-y-time oscilloscope may be
applied torque (Note 2). For Device Type 2, the torque is
used for this purpose). The electronic measuring device used
measured at the passive end with a torque transducer, see 6.4.
must have amplifiers with sufficient gain to observe the torque
NOTE 2—Calibrations at more than one frequency may be needed when
motor input and motion transducer outputs over the entire
range of frequencies anticipated. For measurement of damping
by the free-vibration method, and for calibration of the
The boldface numbers in parentheses refer to a list of references at the end of
this standard. apparatus damping, the readout instrument shall be capable of
D4015 − 21
recording the decay of free vibration with appropriate response 7. Test Specimen
time. A digital x-y-time oscilloscope may be used for this
7.1 General—These methods are limited to the special
purpose.
specimen preparation procedures related to the vibration and
6.6.2 For Device Type 2, a dual channel readout device or a
resonant-column technique. Since the resonant-column test
spectrum analyzer must be used to measure the magnitude and
may be conducted in conjunction with controlled static axial
phase (or real and imaginary) components of the measured
andlateralstresses,theprovisionsforpreparationofspecimens
rotation of the active end relative to the applied torque at the
in Test Method D2166/D2166M, D2850,or D4767 may be
passive end.
applicable or may be used as a guide in connection with other
6.7 Support for Vibration Excitation Device—It may be
methods of application and control of static axial and lateral
necessary to support all or a portion of the weight of the
stresses.
active-end platen and excitation device to prevent excessive
7.2 Specimen Size Limitations—Specimens shall be of uni-
axial stress or compressive failure of the specimen. This
form circular cross section with ends perpendicular to the axis
support may be provided by a spring, counterbalance weights,
of the specimen. Specimens shall have a minimum diameter of
or pneumatic device if the supporting system does not prevent
33 mm (1.3 in.). The largest particle contained within the test
axial movement of the active-end platen and if it does not alter
specimen shall be one sixth of the specimen diameter. If, after
the vibration characteristics of the excitation device.
completion of a test, it is found that larger particles than
6.8 Temporary Platen Support Device—Temporary support
permitted are present, indicate this information in the report of
of the active-end platen may be any clamping device that can
test data under “Remarks.” The length-to-diameter ratio shall
be used to support the platen during attachment of vibration
be not less than 2 nor more than 7 except that, when a static
excitation device to prevent specimen disturbance during
axial stress greater than the lateral stress is applied to the
apparatus assembly. This device is to be removed prior to the
specimen, the ratio of length to diameter shall be between 2
application of vibration.
and 3. Take diameter measurements to the nearest 0.25 mm
(0.01 in.), at the third points along the specimen length and
6.9 Specimen Dimension-Measuring Devices—Dimension-
average them. Take height measurements, to the nearest 0.25
measuring devices are needed to measure portions of the
mm (0.01 in.), at four quadrants and average them. For
apparatusduringcalibrationandspecimendiameterandlength.
determination of moisture content (Test Method D2216),
Any suitable device may be used to make these measurements
secure a representative specimen of the cuttings from intact
except that the device(s) used to measure the length and
specimens, or of the extra soil for remolded specimens, placing
diameter of the specimen must not deform or otherwise affect
the specimen immediately in a covered container.
thespecimen.Speciallydesignedperimetertapes thatmeasure
circumference but read out in diameter are preferred for
7.3 End Coupling for Torsion—For torsional motion, com-
measuring specimen diameters. Measurement accuracies are
plete coupling of the ends of the specimen to the specimen cap
specified in 7.2.
and base must be assured. Coupling for torsion may be
6.10 Balances—Devices for determining the mass of the assumedifthemobilizedcoefficientoffrictionbetweentheend
platens and the specimen is less than 0.2 for all shear strain
soil specimens as well as portions of the device during
calibration. All measurements of mass shall be accurate to amplitudes. The coefficient of friction is approximately given
by:
0.1 %. (Guide D4753)
γG
6.11 Specimen Preparation and Triaxial Equipment—These
Mobilized Coefficient of Friction 5 (1)
'
σ
methods cover specimen preparation and procedures related to
a
the vibration of the specimen and do not cover the application
where:
and control of static axial and lateral stresses.Any or all of the
γ = shear strain amplitude (see Calculations section),
apparatus described in Test Method D2166/D2166M, D2850,
G = shear modulus (see Calculations section), and
or D4767 may be used for specimen preparation and applica-
σ' = effective axial stress.
a
tion of static axial and lateral stresses. Additional apparatus
NOTE 4—The shear strain is not in % for this calculation.
may be used for these purposes as needed.
7.3.1 When this criterion is not met, other provisions such
6.12 Auxiliary Equipment—The auxiliary equipment con-
as the use of adhesives or other friction increasing measures
sists of specimen trimming and carving instruments, a mem-
must be made in order to assure complete coupling (2). In such
brane expander, remolding apparatus, and moisture content
cases, the effectiveness of the coupling provisions shall be
containers as required.
evaluated by testing two specimens of the same material but of
different length. The lengths of these specimens shall differ by
The sole source of supply of the apparatus known to the committee at this time
at least a factor of 1.5.The provisions for end coupling may be
is PI Tape, Box 398, Lemon Grove, CA92045 (http://www.pitape.com). If you are
considered satisfactory if the values of the shear modulus for
aware of alternative suppliers, please provide this information to ASTM Interna-
these two specimens of different length do not differ by more
tional Headquarters.Your comments will receive careful consideration at a meeting
of the responsible technical committee, which you may attend. than 10 %.
D4015 − 21
8. Apparatus Properties (see Note 5)
n = number of components attached to active-end platen
NOTE 5—Practice D3740 provides information on calibration intervals,
and not covered in determination of (J ) .
a 1
records, and quality assurance.
The total rotational inertia for the active end is given by:
8.1 Motion Transducers—Motion transducers shall be cali-
J 5 J 1 J (8)
~ ! ~ !
a a a
1 2
brated with an independent method to ensure calibration
accuracy within 5 % and must be traceable to a government
8.2.1 Acceptablealternateproceduresfordetermining J are
a
standards agency.
provided in A2.1.
8.1.1 Rotational Motion Transducer—The rotational motion
8.3 Apparatus Resonant Frequencies, Spring Constants,
at the free end of the soil specimen is normally measured using
andDampingConstants(onlyneededforDeviceType1)—(See
linearmotiontransducer(s)mountedataradialdistance r from
t
Note 6) Apparatus resonant frequencies and spring constants
the axis of rotation. Linear motion transducers that are sensi-
are defined only for Device Type 1 that has springs attached to
tive to acceleration, velocity or displacement may be used.
the active-end platen system. To determine the resonant
Rotational measuring transducers are acceptable as well. (See
frequencies, set up the apparatus complete with active-end
6.6.)
platen and O-rings used to seal the membranes, but with no
8.1.1.1 The rotation transducer sensitivity S in terms of
θ
specimen.Vibrate at low amplitude and adjust the frequency of
millivolts/radian is computed as follows:
vibration until the input torque is in phase with the velocity of
For an accelerometer transducer with sensitivity S [mV/g]
a
the active-end platen system. This apparatus resonant fre-
the rotation transducer sensitivity at frequency f [Hz] is:
quency is f . The apparatus spring constant, k , is calculated
a a
S 5 S r 2 π f 1 ⁄ 9.81 (2)
~ ! ~ ! from:
θ a t
k 5 ~2 π f ! J (9)
For a velocity transducer with sensitivity S [mV/(m/s) ] the
a a a
v
rotation transducer sensitivity is:
where J is defined in the previous subsection.
a
S 5 S r 2 π f (3)
~ !
θ v t
NOTE 6—Device Type 2 apparatus may or may not have springs and
dashpots attached to the active end platen but by Eq A1.3, these and the
ForadisplacementtransducerwithsensitivityS [mV/m]the
d
active end platen inertia do not affect the determination of shear modulus
rotation transducer sensitivity is:
and damping of the soil.
S 5 S r (4)
θ d t
8.3.1 Apparatus Damping Coefficient for Device Type 1
Rotation of the top of the specimen is given by: apparatus without springs attached to the active end platen.
Device Type 1 without springs may still have a damping
RTrdg@mV#
θ rad 5 (5)
@ # constanttoaccountforbackEMF,aerodynamicdrag,vibration
mV
S
F G of wires attached to the platen, and eddy currents. To measure
θ
rad
the damping constants for the apparatus, attach the same
where RTrdg is the output of the rotation transducer.
masses as used for the determination of apparatus resonant
frequencies. For apparatus without springs attached to the
8.2 Active-End Rotational Inertia (only needed for Device
active-end platen, insert the calibration rod described in the
Type 1)—The rotational inertia, J , of the active-end platen
a
previous subsection. Vibrate the system at the resonant fre-
shall be determined with all transducers and rigid attachments,
quency and measure the torque and rotational motion. The
including attached portions of the vibration excitation device,
apparatus damping coefficient is given by:
securely in place. The rotational inertia of the concentric solid
cylindrical components of the active-end platen and its attach- τ τ τ ω
appl appl appl
c 5 5 5 (10)
a 2
ments is computed from: θω dθ d θ
dt dt
n
J 5 M d (6)
~ !
a ( i i
i51
where:
τ = amplitude of applied torque,
appl
where:
th θ = amplitude of rotation,
M = mass of i solid cylindrical component,
i
dθ
th
d = diameter of i solid cylindrical component, and
= amplitude of rotational velocity,
i
dt
n = number of solid cylindrical components.
d θ
= amplitude of rotational acceleration, and
Transducers and other masses attached to this platen can be dt
ω = resonant circular frequency of the system at calibration
accounted for by:
(=2πf).
n
J 5 J 1 M r (7)
~ ! ~ ! 8.3.2 An acceptable alternate method for calculating the
a 2 ( i i i
i51
apparatus damping coefficient, c is given in A2.2. Reference
a
where:
(3) provides a convenient method for determining both J and
a
th
c that makes use of the program given in Appendix X1.
J = rotational inertia of the i component,
a
i
th
M = mass of i component,
i
8.4 Torque Motor Torque/Current Characteristics (only
th
r = distance from the platen axis to center of mass for i
i
needed for Device Type 1)—For Device Type 1 apparatus
component, and
without springs attached to the active-end platen, insert the
D4015 − 21
calibration rod as described earlier. For Device Type 1 appa- connected to the passive end platen and provide the basis for
ratus with springs attached, set up the apparatus complete with the passive end rotational inertia:
active-end platen and O-rings but no specimen. For either
J 5 J 1J (14)
p passive platen sens head
setup, determine the resonant frequency of this single-degree-
where:
of-freedom system consisting of the active-end platen and
J = calculated using Eq 6-8, and
apparatus spring (or calibration rod) by use of the same
passive platen
J = frequently is provided by the transducer
procedure as described later in the procedures section.Then set sens head
manufacturer.
the frequency to 0.707 times the resonant frequency and apply
torque so that the vibration transducer output to the readout
8.5.2 Alternative methods provided in A2.3.
device has a signal of at least ten times the signal due to
8.5.3 The torque transducer sensitivity is given by the
ambient vibrations and electrical noise when no torque is
manufacturer and must be traceable to a government standards
applied. Read and record the output of the vibration transducer
agency. The torque measured by the torque transducer is
and the current input to the torque generating instrument
calculated from:
(torque motor). Next, set the frequency to 1.414 times the
TT
rdg
system resonant frequency and obtain the readings similar to
τ 5 (15)
TT
TT
sens
those at 0.707 times the resonant frequency. Calculate C and
where:
C from:
TT = Torque Transducer sensitivity typically in units of
θ sens
C 5 (11)
1 mV/(N-m)
2CR
TT = Voltage reading (mV) for the torque transducer.
rdg
θ
C 5 9. Procedure
CR
9.1 Test Setup—The exact procedure to be followed during
where:
test setup will depend on the apparatus and electronic equip-
θ = active-end rotation at 0.707 times resonant frequency
ment used and on methods used for application, measurement,
(Note 7),
and control of the static axial and lateral stresses. However, the
CR = torque motor input (amps) at 0.707 times resonant
specimen shall be placed in the apparatus by procedures that
frequency (Note 8),
will minimize the disturbance of the specimen. Particular care
θ = active-end transducer output at 1.414 times resonant
must be exercised when attaching the end platens to the
frequency (Note 7), and
specimen and when attaching the vibration excitation device to
CR = torque motor input (amps) at 1.414 times resonant
the platens. A temporary support as discussed earlier may be
frequency (Note 8).
needed. For cases where isotropic static stresses are to be
NOTE 7—θ and θ will be functions of frequency for velocity and
1 2
applied to a membrane-enclosed specimen, liquid- or air-
acceleration measuring transducers (see 8.1).
confining media may be used for dry or partially saturated
NOTE 8—If a current-measuring instrument is used, the units will be
specimens. For tests where complete saturation is important, a
amperes.Alternatively, voltage drop across a fixed resistance may also be
liquid-confining medium shall be used. Where the vibration
measured and the units will then be volts.
excitation device is located within the pressure chamber, an
By use of C and C , the torque motor rating, TMR,is
1 2
air-liquid interface is acceptable if the liquid covers the entire
obtained from:
membrane that encloses the specimen.
TMR 5 0.5k C 1 C (12)
~ !
1 2
9.2 Electronic Equipment—Turn off the power supplied to
where: the torque motor. Connect the torque motor to the sine wave
generator (with amplifier, if required). Connect the vibration
k = apparatus spring constant, k (or for apparatus without
a
transducers to the readout instruments. Gradually apply power
springs, the calibrating rod spring constant, k ).
rod
to the torque motor and adjust the readout instruments accord-
The torque applied to the top platen by the torque generator
ing to the instruction manuals for these instruments.
is given by:
9.3 Measurements:
τ 5TMR·T (13)
appl rdg
9.3.1 Device Type 1:
9.3.1.1 Measurement of Resonant Frequency—The motion
where:
of the active-end platen in conjunction with the applied torque
T = input amps to the torque motor
rdg
is used to establish resonance. Resonance is defined as the
TMR = torque motor rating from Eq 12.
lowest frequency where the torque is 90 degrees out-of-phase
8.5 Passive End Inertia and Torque Transducer Calibra-
with the rotational acceleration or displacement. This phase
tions (only needed for Device Type 2):
relationship can be detected by observing the Lissajous figure
8.5.1 A torque transducer generally consists of a metallic on an oscilloscope with the torque input signal and rotational
case containing a “spring” instrumented to measure strain acceleration or displacement plotted as x-y.(Note 9)At the 90
where the strain is proportional to the applied torque. The degree phase relationship the figure will be an ellipse with its
torque is applied to the spring through a sensing head protrud- axes vertical and horizontal. If a velocity transducer is used for
ing from the transducer case. The sensing head must be rigidly rotational measurement, the system resonance occurs when the
D4015 − 21
Lissajous figure forms a straight, sloping line. It is recom- resonant frequencies. Thus, for a given torque, the vibration
mended that the frequency be measured with a digital elec- motion transducer outputs recorded at the resonant frequency
tronic frequency meter and be recorded to at least three give sufficient information to calculate strain amplitude. To
significant figures. increase or decrease strain amplitude, the applied torque must
beincreasedordecreased.Aftermakingachangeintorque,the
9.3.1.2 The determination of the lowest resonant frequency
procedure of 9.3.2.1 must be followed to establish the corre-
can be done by setting the torque excitation frequency (for
sponding resonant frequency before the rotation transducer
example,10Hz)andpowertoaslowavalueaspractical.Then
output can be used to establish the new shear strain amplitude
increase the frequency of excitation until the system resonant
value.
frequency is obtained.
9.3.2.3 Measurement of System Damping—Damping is de-
NOTE 9—The phase relationship between two signals may also be
termined from steady-state measurements of torque measured
computed by measurement of the time difference between zero crossings
at the base of the specimen (passive end), amplitude of motion
of the two signals divided by the period of the oscillations (period =
of the active end and the phase difference between them as
) multiplied by 360 gives the phase in degrees. If the signals are not
frequency
clean sine waves, then a spectral analysis will have to be performed to get
described in the next section.
accurate values for magnitude and phase (or real and imaginary compo-
nents) of the rotation/torque ratio. The magnitude of the rotation/torque
10. Calculation
ratio multiplied by the cosine of the phase gives the real component of the
rotation/torque ratio and the same ratio multiplied by the sine of the phase
10.1 General—Calculations require the apparatus calibra-
gives the imaginary component of the rotation/torque ratio.
tion factors and the physical dimensions and mass of the
9.3.1.3 Measurement of Strain—The strain amplitude mea- specimen at the time resonant measurements are made. In
surements shall be made only at the system resonant frequen-
addition, for each static axial and lateral stress condition, one
cies. Thus, for a given torque, the vibration motion transducer data set shall be measured for each vibration strain amplitude.
outputs recorded at the system resonant frequency give suffi-
A data set consists of: duration of vibration (this time can be
cient information to calculate strain amplitude. To increase or used to calculate the number of vibration cycles), system
decrease strain amplitude, the applied torque must be increased
resonant frequency, active-end transducer output for both type
ordecreased.Aftermakingachangeintorque,theprocedureof devices. For DeviceType 1 additionally, the reading associated
9.3.1.1 must be followed to establish the corresponding system with the applied torque, and if the amplitude decay method of
resonantfrequencybeforetherotationtransduceroutputcanbe measuring damping is also going to be used, the free-vibration
used to establish the new shear strain amplitude value. amplitude decay curve. For Device Type 2, it is necessary to
measure the torque as well as the phase between the torque
9.3.1.4 Measurement of System Damping—Associated with
transducer output and the motion at the active end of the
each shear strain amplitude and system resonant frequency is a
specimen (Note 9).
value of damping. Two methods are available for measuring
system damping: the steady-state vibration method and the 10.1.1 The calculations outlined in this section may all be
madebycomputerprograms.ForDeviceType1,aprogramfor
amplitude decay method. Both methods should give similar
results. The steady-state method is easier and quicker. It is making the calculations is provided in Appendix X1. For
Device Type 2, the program is given in Appendix X2. Other
generally always used and the amplitude decay method is used
for occasional spot-checking. For the steady-state method, the programs may be used to make a portion or all of the
active-end transducer output and the applied torque must be calculations as long as they provide identical results. The units
measured at each resonant frequency. The calculations are for the symbols in this section are given in Annex A3.
outlined in the following section. For the free-vibration
10.2 Soil Mass Density—The soil mass density, ρ, is given
method, with the system vibrating at the system resonant
by:
frequency, cut the power to the vibration excitation device and
M
record the output of the rotation transducer used in establishing
ρ 5 (16)
V
resonance as a function of time. The shut-off mechanism must
create an open circuit with the vibration excitation device and
where:
cannot be done by switching off the power to amplifier.
M = total mass of specimen, and
Without an open circuit, damping will be induced by current
V = volume of specimen.
flow in the circuit. This gives the decay curve for free
10.3 Specimen Rotational Inertia—The specimen rotational
vibration. The calculations for damping are outlined in the
inertia about the axis of rotation is given by:
following section.
Md
9.3.2 Device Type 2:
J 5 (17)
9.3.2.1 Measurement of Resonant Frequency—This is the
lowest frequency at which the active end rotation is a maxi-
where d = diameter of specimen.
mum. In addition to measuring the frequency, magnitude of
10.4 Active-End Inertia Factors:
motion and magnitude of torque, the phase between the motion
10.4.1 The active-end inertia factor, T , is only needed for
at the active end and the torque at the passive end must be a
Device Type 1 and is given by:
determined (see Note 9).
J f
9.3.2.2 Measurement of Strain Amplitude—The strain am-
a a
T 5 1 2 (18)
F S D G
a
plitude measurements shall be made only at the system J f
r
D4015 − 21
where: 10.7.1.2 The dimensionless frequency, λ* is determined by
* *
setting DSS from Eq 22 equal to DSS from Eq 23
J = rotational inertia of active-end platen system as calcu- meas calc
a
using an optimization routine. The computer program in
lated earlier,
Appendix X1, which is written in Excel, solves for λ*by
J = specimen rotational inertia as calculated earlier,
comparing Eq 23 the results with Eq 22.
f = apparatus resonant frequency (for apparatus without
a
springs attached to the active-end platen, this factor is
10.7.1.3 For Device Type 2, multiplying Eq A1.3 by ω J
zero), and
and with the assumption that ωc << k gives:
p p
f = system resonant frequency.
r 2 2
J ω ω
* * *
~MMF ! 5 cosλ 1 1 2 λ sinλ (24)
S D F S D G
calc
DT2
10.5 Apparatus Damping Factors:
J ω ω
p p p
10.5.1 The apparatus damping factor, for Device Type 1 is
where λ* is defined by Eq A1.4 and
calculated from:
k
c
p
a
ω 5Œ (25)
ADF 5 (19)
p
a
J
2πf J
p
r
where c = apparatus damping coefficient as described by Eq 10.7.2 Dimensionless Frequency Factor—The dimension-
a
*
10 or A2.2. less frequency factor, λ , is complex having both a real
component,λ ,andanimaginarycomponent,λ .Itisusedin
Re Im
10.6 Modified Magnification Factor:
calculating modulus and damping by solving Eq 22 and Eq 23
10.6.1 The measure
...

This document is not an ASTM standard and is intended only to provide the user of an ASTM standard an indication of what changes have been made to the previous version. Because
it may not be technically possible to adequately depict all changes accurately, ASTM recommends that users consult prior editions as appropriate. In all cases only the current version
of the standard as published by ASTM is to be considered the official document.
´1
Designation: D4015 − 15 D4015 − 21
Standard Test Methods for
Modulus and Damping of Soils by Fixed-Base Resonant
Column Devices
This standard is issued under the fixed designation D4015; 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.
ε NOTE—Editorially corrected Eq 28 in February 2017.
1. Scope*
1.1 These test methods cover the determination of shear modulus and shear damping as a function of shear strain amplitude for
solid cylindrical specimens of soil in intact and remoldedreconstituted conditions by torsional vibration using resonant column
devices. The vibration of the specimen may be superposed on a controlled static state of stress in the specimen. The vibration
apparatus and specimen may be enclosed in a triaxial chamber and subjected to an all-around pressure and axial load. In addition,
the specimen may be subjected to other controlled conditions (for example, pore-water pressure, degree of saturation, temperature).
These test methods of modulus and damping determination are considered nondestructive when the shear strain amplitudes of
–2 –4
vibration are less than 10 % (10 in. ⁄in.), and many measurements may be made on the same specimen and with various states
of static stress.
1.2 Two device configurations are covered by these test methods: Device Type 1 where a known torque is applied to the top of
the specimen and the resulting rotational motion is measured at the top of the specimen, and Device Type 2 where an uncalibrated
torque is applied to the top of the specimen and the torque transmitted through the specimen is measured by a torque transducer
at the base of the specimen. For both types of devices, the torque is applied to the active end (usually top) of the specimen and
the rotational motion also is measured at the active end of the specimen.
1.3 These test methods are limited to the determination of the shear modulus and shear damping, the necessary vibration, and
specimen preparation procedures related to the vibration, etc., and do not cover the application, measurement, or control of the
axial and lateral static normal stresses. The latter procedures may be covered by, but are not limited to, Test Method D2850,
D3999/D3999M, D4767, D5311/D5311M, or D7181.
1.4 Significant Digits—All recorded and calculated values shall conform to the guide for significant digits and rounding
established in Practice D6026.
1.4.1 The procedures used to specify how data are collected/recorded and calculated in this standard are regarded as the industry
standard. In addition, they are representative of the significant digits that should generally be retained. The procedures used do not
consider material variation, purpose for obtaining the data, special purpose studies, or any considerations for the user’s objectives;
and it is common practice to increase or reduce significant digits of reported data to be commensurate with these considerations.
It is beyond the scope of this standard to consider significant digits used in analysis methods for engineering design.
These test methods are under the jurisdiction of ASTM Committee D18 on Soil and Rock and are the direct responsibility of Subcommittee D18.09 on Cyclic and
Dynamic Properties of Soils.
Current edition approved Oct. 1, 2015Dec. 1, 2021. Published November 2015December 2021. Originally approved in 1981. Last previous edition approved in 20072015
ɛ1
as D4015 – 07.D4015 –15 . DOI: 10.1520/D4015-15E01.10.1520/D4015-21.
*A Summary of Changes section appears at the end of this standard
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
D4015 − 21
1.4.2 Measurements made to more significant digits or better sensitivity than specified in this standard shall not be regarded a
nonconformance with this standard.
1.5 Units—The values stated in SI units are to be regarded as standard. The values given in parentheses are mathematical
conversions to inch-pound units, which are provided for information only and are not considered standard. Reporting of test results
in units other than SI shall not be regarded as nonconformance with these test methods.
1.5.1 The converted inch-pound units use the gravitational system of units. In this system, the pound (lbf) represents a unit of force
(weight), while the unit for mass is slugs. The converted slug unit is not given, unless dynamic (F = ma) calculations are involved.
1.5.2 It is common practice in the engineering/construction profession to concurrently use pounds to represent both a unit of mass
(lbm) and of force (lbf). This implicitly combines two separate systems of units; that is, the absolute system and the gravitational
system. It is scientifically undesirable to combine the use of two separate sets of inch-pound units within a single standard. As
stated, this standard includes the gravitational system of inch-pound units and does not use/present the slug unit for mass. However,
the use of balances or scales recording pounds of mass (lbm) or recording density in lbm/ft shall not be regarded as
nonconformance with this standard.
1.6 This standard does not purport to address all of the safety concerns, if any, associated with its use. It is the responsibility
of the user of this standard to establish appropriate safety safety, health, and healthenvironmental practices and determine the
applicability of regulatory limitations prior to use.
1.7 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.
2. Referenced Documents
2.1 ASTM Standards:
D653 Terminology Relating to Soil, Rock, and Contained Fluids
D2166/D2166M Test Method for Unconfined Compressive Strength of Cohesive Soil
D2216 Test Methods for Laboratory Determination of Water (Moisture) Content of Soil and Rock by Mass
D2850 Test Method for Unconsolidated-Undrained Triaxial Compression Test on Cohesive Soils
D3740 Practice for Minimum Requirements for Agencies Engaged in Testing and/or Inspection of Soil and Rock as Used in
Engineering Design and Construction
D3999/D3999M Test Methods for the Determination of the Modulus and Damping Properties of Soils Using the Cyclic Triaxial
Apparatus (Withdrawn 2020)
D4753 Guide for Evaluating, Selecting, and Specifying Balances and Standard Masses for Use in Soil, Rock, and Construction
Materials Testing
D4767 Test Method for Consolidated Undrained Triaxial Compression Test for Cohesive Soils
D5311/D5311M Test Method for Load Controlled Cyclic Triaxial Strength of Soil
D6026 Practice for Using Significant Digits and Data Records in Geotechnical Data
D7181 Test Method for Consolidated Drained Triaxial Compression Test for Soils
3. Terminology
3.1 Definitions—For definitions of other common technical terms used in these test methods, seethis standard, refer to
Terminology D653.
3.2 Definitions of Terms Specific to This Standard:
3.2.1 damping capacity D [unitless, typically expressed in %],n—in resonant column systems, is related to the component of the
dynamic shear modulus that lags the applied shear stress by 90° degrees.
3.2.2 Device Type 1, DT1, n—in resonant column systems, a resonant column system as shown in Fig. 1 where the passive end
platen is directly connected to the Fixed Base (no torque transducer), a calibrated vibratory torque is applied to the active end, and
rotation is measured at the active end.
For referenced ASTM standards, visit the ASTM website, www.astm.org, or contact ASTM Customer Service at [e-mail]. For Annual Book of ASTM Standards
volume information, refer to the standard’s Document Summary page on the ASTM website.
D4015 − 21
For Device Type 1, no torque transducer is needed and the Passive End Platen is connected to the Fixed Base.
FIG. 1 Resonant-Column Schematic for Both Device Types 1 and 2
3.2.2.1 Discussion—
The vibration excitation device may incorporate springs and dashpots connected to the active-end platen, where the spring
constants and viscous damping coefficients must be known. The rotational inertia of the active-end platen and portions of the
vibration excitation device moving with it must be known.
3.2.3 Device Type 2, DT2, n—in resonant column systems, a resonant column system as shown in Fig. 1 where the passive end
platen is connected to a torque transducer, an uncalibrated torque is applied to the active end, torque is measured by the torque
transducer at the passive end, and rotation is measured at the active end.
3.2.3.1 Discussion—
The vibration excitation device may incorporate springs and dashpots connected to the active-end platen, but the spring constants
and viscous damping coefficients are not needed. The rotational inertia of the active-end platen and portions of the vibration
excitation device moving with it also are not needed.
3.2.4 dimensionless specimen stiffness, DSS* [unitless], n—in resonant column systems, is a complex number used to characterize
the real and imaginary components of the specimen stiffness.
-2
3.2.5 dynamic shear modulus, G* [FL ], n—in resonant column systems, is the ratio of shear stress to shear strain under vibratory
conditions (also known as complex shear modulus).
-2
3.2.6 equivalent elastic shear modulus G [FL ], n—in resonant column systems, is the component of the dynamic shear modulus
that is in-phase with the applied shear stress.
3.2.7 resonant-column system, n—a system as shown in Fig. 1 consisting of a cylindrical specimen or column of soil enclosed with
a flexible membrane that has platens attached to each end and where a sinusoidal vibration excitation device is attached to the
active-end platen and where the other end is the passive-end platen that is rigidly fixed.
3.2.8 specimen shear strain γ, [unitless, frequently expressed as %],n—in resonant column systems, is the average shear strain in
the specimen where the shear strain in each cross section varies from zero along the axis of rotation to a maximum at the perimeter
of the specimen.
D4015 − 21
3.2.8.1 Discussion—
The radius for calculating average shear strains vary depending on soil type, strain level, confining stress, etc. The default value
of the radius for calculating average strain is 0.4*diameter but values in the range of 0.33 to 0.40*diameter may be used if the value
is documented in the report.
-1
3.2.9 system resonant frequency f [s ],n—in resonant column systems, for Device Type 1 is the lowest frequency at which the
r
rotational velocity at the active end is in phase with the sinusoidal excitation torque and for Device Type 2, is the lowest frequency
at which the rotational motion at the active end is a maximum.
4. Summary of Test Method
4.1 The resonant column device is shown schematically in Fig. 1. In the resonant column test, a cylindrical soil specimen, usually
enclosed with a thin membrane, is subjected to an imposed static axial and lateral stress condition. Torsional sinusoidal vibrations
are applied at the top of the soil specimen and the rotational response is measured. The frequency of excitation is varied until the
system resonant frequency is achieved as described in 3.2.83.2.9. The devices may be operated at frequencies other than resonant
frequencies. Given the geometry, mass and system parameters, the equivalent elastic shear modulus and damping capacity can be
determined at a measured level of excitation vibration. The amplitude of vibration (which is related to shear strain) is typically
varied to measure the variation of modulus and damping as a function of shear strain. The test is usually conducted at levels of
shear strain between 0.00001 % and 0.2 %. (The upper limit of shear strain is dependent on the specimen stiffness and the
maximum torque capability of the excitation system.) For specimens where the maximum shear strain measured is of the order of
0.01 %, the test is often conducted at several different sets of static axial and lateral stress conditions to measure the variation of
moduli and damping with static stress states. The test results are dependent on sample quality/specimen disturbance which are
beyond the scope of this standard.
5. Significance and Use
5.1 The equivalent elastic shear modulus and damping capacity of a given soil, as measured by the resonant column technique
herein described, depend upon the strain amplitude of vibration, the state of effective stress, and the void ratio of the soil,
temperature, time, etc. Since the application and control of the static axial and lateral stresses and the void ratio are not prescribed
in these methods, the applicability of the results to field conditions will depend on the degree to which the application and control
of the static axial and lateral stresses and the void ratio, as well as other parameters such as soil structure, duplicate field conditions.
The techniques used to simulate field conditions depend on many factors and it is up to the engineer to decide on which techniques
apply to a given situation and soil type. The results of these tests are useful for calculations involving soil-structure interaction and
seismic response of soil deposits.
NOTE 1—The quality of the results produced by this standard is dependent on the competence of the personnel performing it, and the suitability of the
equipment and facilities. Agencies that meet the criteria of Practice D3740 are generally considered capable of competent and objective
testing/sampling/inspection/etc. Users of this standard are cautioned that compliance with Practice D3740 does not in itself assure reliable results. Reliable
results depend on many factors; Practice D3740 provides a means of evaluating some of those factors.
6. Apparatus
6.1 General—The complete test apparatus is shown schematically in Fig. 1 and includes the platens for holding the specimen in
the pressure cell, the vibration excitation device (torque motor) , motor), transducers for measuring the response, the control and
readout instrumentation, and auxiliary equipment for specimen preparation. The theory for the resonant column is provided in
Annex A1. The entire apparatus is generally enclosed within a pressure chamber (commonly referred to as a triaxial cell). For some
apparatus that can apply an axial load to the specimen, the pressure chamber lid may be fitted with a piston passing through the
top.
6.2 Specimen Platens—Both the active-end and passive-end platens shall be constructed of noncorrosive material having a
modulus at least ten times the modulus of the material to be tested. Each platen shall have a circular cross section and a plane
surface of contact with the specimen, except that the plane surface of contact mayshall be roughened to provide for more efficient
coupling with the ends of the specimen. Roughening and flow of fluids into or from the specimen mayshall be accomplished by
rigidly fastening porous disks to the platens. The diameter of platens shall be equal to or greater than the diameter of the specimen.
The construction of the platens shall be such that their stiffness is at least ten times the stiffness of the specimen.
6.2.1 The active-end platen may have a portion of the excitation device, transducers, springs, and dashpots connected to it. The
transducers and moving portions of the excitation device must be connected to the platen in such a fashion that they are to be
D4015 − 21
considered part of the platen, be counterbalanced to maintain rotational symmetry, and have the same motion as the platen for the
full range of frequencies to be encountered when testing soils.
6.2.2 The theoretical model used for the resonant-column system represents the active-end platen, with all attachments, as a rigid
mass that is attached to the specimen; this mass may also have massless springs and dashpots attached to it as shown in Fig. 1.
If springs are used, the excitation device and active-end platen (without the specimen in place) form a one-degree-of-
freedomsingle-degree-of-freedom system having an undamped natural frequency, f .
a
6.2.3 The passive-end platen must be rigidly fixed. It may be assumed to be rigidly fixed when the inertia of it and the mass(es)
attached to it are at least 500 times the inertia of the active-end platen (1) .
6.2.4 For Device Type 2, a torque transducer is placed between the passive end platen and the rigidly fixed base. The torque
transducer even though relatively stiff in torsion (see 6.4), must allow for some small rotation of the passive end platen in order
to register the transmitted torque. The inertia of the passive end platen system, J must include the inertia of the sensing head of
p
the torque transducer which is rigidly fastened to it. With no specimen in place, the passive end platen system inertia, J , along
p
with the stiffness, k , and damping coefficient, c , of the torque transducer constitute a single-degree-of-freedom system which are
p p
accounted for in Eq A1.3.
6.3 Vibration Excitation Device (torque motor)—This shall be a device capable of applying a sinusoidal torsional vibration to the
active-end platen to which the moving parts of the device are rigidly coupled. The frequency of excitation shall be continuously
variable and have a range that typically includes 10 Hz to 1 kHz. For Device Type 1 where the torque is measured at the active
end, the excitation device shall have a means of measuring the torque applied to the excitation device that has at least 5 % accuracy
of full scale full-scale output. If an electromagnetic excitation device is used, the voltage drop across a fixed, temperature-and-
frequency-stable power resistor in series with the excitation device is proportional to applied torque (Note 2). For Device Type 2,
the torque is measured at the passive end with a torque transducer, see 6.4.
NOTE 2—Calibrations at more than one frequency may be needed when testing frequencies vary over a wide range. Use of several calibration rods with
differing torsional stiffness may be needed.
6.4 Passive End Torque Transducer—This torque transducer for Device Type 2 must be waterproof and insensitive to ambient
pressure and temperature changes for the expected values. It may be a transducer that also measures axial force. The torque
transducer must have a torque capacity of at least twice the maximum torque capability of the vibration excitation device, a
linearity of 60.5 % of full scale full-scale output, hysteresis less than 60.1 % of full scale full-scale output, and repeatability better
than 60.5 % of full scale full-scale output. If the transducer is used to measure axial force, the specifications must be similar to
those for torque. The transducer must be rigidly connected to the chamber base and the sensing head of the torque transducer shall
be rigidly connected to the passive end platen.
6.5 Sine Wave Generator—The sine wave generator is an electricelectronic instrument capable of producing a sinusoidal current
with a means of adjusting the frequency over the entire range of operating frequencies anticipated. This instrument shall provide
sufficient power to produce the desired vibration amplitude, or its output may be electronically amplified to provide sufficient
power.
6.6 Vibration-Measuring Devices and Readout Instruments—These devices and instruments shall be calibrated with an accuracy
of 5 %5 % and must be traceable to a government standards agency. The vibration-measuring devices shall be acceleration,
velocity, or displacement transducers that can be attached to and become a part of the active-end platen. The transducer(s) shall
be mounted to produce a calibrated electrical output that is proportional to the rotational acceleration, velocity, or displacement.
If the rotation is measured using linear motion measurement transducers acting at a radius from the axis of rotation, two transducers
must be used with both at the same radius from the axis of rotation but connected diagonally across the axis of rotation and they
must be wired such that rotational motion is additive and lateral translation is subtracted (Note 3). The readout instruments must
have a frequency resolution of at least 0.1 Hz. It also is necessary to have an electronic device for establishing the phase difference
between the applied and/or measured torque and resulting rotational motion for establishing the system resonant frequency.motion.
NOTE 3—The use of two linear motion transducers used to measure rotation will minimize resonances in the bending modes from being confused with
those for the torsional modes.
The last approved version of this historical standard is referenced on www.astm.org.
The boldface numbers in parentheses refer to a list of references at the end of this standard.
D4015 − 21
6.6.1 For Device Type 1, an two types of tests are commonly used: 1) steady-state vibration where applied frequency is varied.
(A dual channel digital oscilloscope, readout device, or spectrum analyzer may be used.) and 2) transient vibration starting from
a steady-state vibration at the resonant frequency and excitation is then cut off and the decay of vibratory motion is measured with
time (a digital x-y-time oscilloscope may be used for this purpose.purpose). The electronic measuring device used must have
amplifiers with sufficient gain to observe the torque motor input and motion transducer outputs over the entire range of frequencies
anticipated. For measurement of damping by the free-vibration method, and for calibration of the apparatus damping, the readout
instrument shall be capable of recording the decay of free vibration with appropriate response time. A digital x-y-time oscilloscope
may be used for this purpose. For Device Type 2, a dual channel readout device or a spectrum analyzer must be used to measure
the magnitude and phase (or real and imaginary) components of the measured θ /τ at the resonant frequency.
a TT
6.6.2 For Device Type 2, a dual channel readout device or a spectrum analyzer must be used to measure the magnitude and phase
(or real and imaginary) components of the measured rotation of the active end relative to the applied torque at the passive end.
6.7 Support for Vibration Excitation Device—It may be necessary to support all or a portion of the weight of the active-end platen
and excitation device to prevent excessive axial stress or compressive failure of the specimen. This support may be provided by
a spring, counterbalance weights, or pneumatic device as long as if the supporting system does not prevent axial movement of the
active-end platen and as long as if it does not alter the vibration characteristics of the excitation device.
6.8 Temporary Platen Support Device—Temporary support of the active-end platen may be any clamping device that can be used
to support the platen during attachment of vibration excitation device to prevent specimen disturbance during apparatus assembly.
This device is to be removed prior to the application of vibration.
6.9 Specimen Dimension-Measuring Devices—Dimension-measuring devices are needed to measure portions of the apparatus
during calibration and specimen diameter and length. Any suitable device may be used to make these measurements except that
the device(s) used to measure the length and diameter of the specimen must not deform or otherwise affect the specimen. Specially
designed perimeter tapes that measure circumference but read out in diameter are preferred for measuring specimen diameters.
Measurement accuracies are specified in 7.2.
6.10 Balances—Devices for determining the mass of the soil specimens as well as portions of the device during calibration. All
measurements of mass shouldshall be accurate to 0.1 %. (Guide D4753)
6.11 Specimen Preparation and Triaxial Equipment—These methods cover specimen preparation and procedures related to the
vibration of the specimen and do not cover the application and control of static axial and lateral stresses. Any or all of the apparatus
described in Test Method D2166/D2166M, D2850, or D4767 may be used for specimen preparation and application of static axial
and lateral stresses. Additional apparatus may be used for these purposes as needed.
6.12 Miscellaneous Apparatus—Auxiliary Equipment—The miscellaneous apparatusauxiliary equipment consists of specimen
trimming and carving tools,instruments, a membrane expander, remolding apparatus, and moisture content canscontainers as
required.
7. Test Specimen
7.1 General—These methods are limited to the special specimen preparation procedures related to the vibration and
resonant-column technique. Since the resonant-column test may be conducted in conjunction with controlled static axial and lateral
stresses, the provisions for preparation of specimens in Test Method D2166/D2166M, D2850, or D4767 may be applicable or may
be used as a guide in connection with other methods of application and control of static axial and lateral stresses.
7.2 Specimen Size Limitations—Specimens shall be of uniform circular cross section with ends perpendicular to the axis of the
specimen. Specimens shall have a minimum diameter of 33 mm (1.3 in.). The largest particle contained within the test specimen
shall be one sixth of the specimen diameter. If, after completion of a test, it is found that larger particles than permitted are present,
indicate this information in the report of test data under “Remarks.” The length-to-diameter ratio shall be not less than 2 ornor more
The sole source of supply of the apparatus known to the committee at this time is PI Tape, Box 398, Lemon Grove, CA 92045 (http://www.pitape.com). If you are aware
of alternative suppliers, please provide this information to ASTM International Headquarters. Your comments will receive careful consideration at a meeting of the responsible
technical committee, which you may attend.
D4015 − 21
than 7 except that, when a static axial stress greater than the lateral stress is applied to the specimen, the ratio of length to diameter
shall be between 2 and 3. Take diameter measurements to the nearest 0.25 mm (0.01 in.), at the third points along the specimen
length and average them. Take height measurements, to the nearest 0.25 mm (0.01 in.), at four quadrants and average them. For
determination of moisture content (Test Method D2216), secure a representative specimen of the cuttings from intact specimens,
or of the extra soil for remolded specimens, placing the specimen immediately in a covered container.
7.3 End Coupling for Torsion—For torsional motion, complete coupling of the ends of the specimen to the specimen cap and base
must be assured. Coupling for torsion may be assumed if the mobilized coefficient of friction between the end platens and the
specimen is less than 0.2 for all shear strain amplitudes. The coefficient of friction is approximately given by:
γG
Mobilized Coefficient of Friction 5 (1)
'
σ
a
where:
γ = shear strain amplitude (see Calculations section),
G = shear modulus (see Calculations section), and
σ' = effective axial stress.
a
NOTE 4—The shear strain is not in % for this calculation.
7.3.1 When this criterion is not met, other provisions such as the use of adhesives or other friction increasing measures must be
made in order to assure complete coupling (2). In such cases, the effectiveness of the coupling provisions shall be evaluated by
testing two specimens of the same material but of different length. The lengths of these specimens shall differ by at least a factor
of 1.5. The provisions for end coupling may be considered satisfactory if the values of the shear modulus for these two specimens
of different length do not differ by more than 10 %.
8. Apparatus Properties (see Note 45)
NOTE 5—Practice D3740 provides information on calibration intervals, records, and quality assurance.
8.1 Motion Transducers—Motion transducers shall be calibrated with an independent method to ensure calibration accuracy within
5 % and must be traceable to a government standards agency.
8.1.1 Rotational Motion Transducer—The rotational motion at the free end of the soil specimen is normally measured using linear
motion transducer(s) mounted at a radial distance r from the axis of rotation. Linear motion transducers that are sensitive to
t
acceleration, velocity or displacement may be used. Rotational measuring transducers are acceptable as well. (See 6.6.)
8.1.1.1 The rotation transducer sensitivity S in terms of millivolts/radian is computed as follows:
θ
For an accelerometer transducer with sensitivity S [mV/g] the rotation transducer sensitivity at frequency f [Hz] is:
a
S 5 S r ~2 π f! ~1 ⁄ 9.81! (2)
θ a t
For a velocity transducer with sensitivity S [mV/(m/s) ] the rotation transducer sensitivity is:
v
S 5 S r ~2 π f! (3)
θ v t
For a displacement transducer with sensitivity S [mV/m] the rotation transducer sensitivity is:
d
S 5 S r (4)
θ d t
Rotation of the top of the specimen is given by:
RTrdg mV
@ #
θ rad 5 (5)
@ #
mV
S
F G
θ
rad
where RTrdg is the output of the rotation transducer.
8.2 Active-End Rotational Inertia (only needed for Device Type 1)—The rotational inertia, J , of the active-end platen shall be
a
determined with all transducers and rigid attachments, including attached portions of the vibration excitation device, securely in
place. The rotational inertia of the concentric solid cylindrical components of the active-end platen and its attachments is computed
from:
D4015 − 21
n
J 5 M d (6)
~ !
a 1 ( i i
i51
where:
th
M = mass of i solid cylindrical component,
i
th
d = diameter of i solid cylindrical component, and
i
n = number of solid cylindrical components.
Transducers and other masses attached to this platen can be accounted for by:
n
J 5 J 1 M r (7)
~ ! ~ !
a i i i
2 (
i51
where:
th
J = rotational inertia of the i component,
i
th
M = mass of i component,
i
th
r = distance from the platen axis to center of mass for i component, and
i
n = number of components attached to active-end platen and not covered in determination of (J ) .
a 1
The total rotational inertia for the active end is given by:
J 5 J 1 J (8)
~ ! ~ !
a a 1 a 2
8.2.1 Acceptable alternate procedures for determining J are provided in A2.1.
a
8.3 Apparatus Resonant Frequencies, Spring Constants, and Damping Constants (only needed for Device Type 1)—(See Note 56)
Apparatus resonant frequencies and spring constants are defined only for Device Type 1 that has springs attached to the active-end
platen system. To determine the resonant frequencies, set up the apparatus complete with active-end platen and O-rings used to
seal the membranes, but with no specimen. Vibrate at low amplitude and adjust the frequency of vibration until the input torque
is in phase with the velocity of the active-end platen system. This apparatus resonant frequency is f . The apparatus spring constant,
a
k , is calculated from:
a
k 5 2 π f J (9)
~ !
a a a
where J is defined in the previous subsection.
a
NOTE 6—Device Type 2 apparatus may or may not have springs and dashpots attached to the active end platen but by Eq A1.3, these and the active end
platen inertia do not affect the determination of shear modulus and damping of the soil.
8.3.1 Apparatus Damping Coefficient for Device Type 1 apparatus without springs attached to the active end platen. Device Type
1 without springs may still have a damping constant to account for back EMF, aerodynamic drag, vibration of wires attached to
the platen, and eddy currents. To measure the damping constants for the apparatus, attach the same masses as used for the
determination of apparatus resonant frequencies. For apparatus without springs attached to the active-end platen, insert the
calibration rod described in the previous subsection. Vibrate the system at the resonant frequency and measure the torque and
rotational motion. The apparatus damping coefficient is given by:
τ τ τ ω
appl appl appl
c 5 5 5 (10)
a
θω dθ d θ
dt dt
where:
τ = amplitude = amplitude of applied torque,
appl
θ = amplitudeθ = amplitude of rotation,
dθ
= amplitude of rotational velocity,
dt
d θ
= amplitude of rotational acceleration, and
dt
ω = resonantω = resonant circular frequency of the system at calibrationcalibration (=2πf).
8.3.2 An acceptable alternate method for calculating the apparatus damping coefficient, c is given in A2.2. Reference (3) provides
a
a convenient method for determining both J and c that makes use of the program given in Appendix X1.
a a
D4015 − 21
8.4 Torque Motor Torque/Current Characteristics (only needed for Device Type 1)—For Device Type 1 apparatus without springs
attached to the active-end platen, insert the calibration rod as described earlier. For Device Type 1 apparatus with springs attached,
set up the apparatus complete with active-end platen and O-rings but no specimen. For either setup, determine the resonant
frequency of this single-degree-of-freedom system consisting of the active-end platen and apparatus spring (or calibration rod) by
use of the same procedure as described later in the procedures section. Then set the frequency to 0.707 times the resonant frequency
and apply torque so that the vibration transducer output to the readout device has a signal of at least ten times the signal due to
ambient vibrations and electrical noise when no torque is applied. Read and record the output of the vibration transducer and the
current input to the torque generating instrument (torque motor). Next, set the frequency to 1.414 times the system resonant
frequency and obtain the readings similar to those at 0.707 times the resonant frequency. Calculate C and C from:
1 2
θ
C 5 (11)
2CR
θ
C 5
CR
where:
θ = active-end rotation at 0.707 times resonant frequency (Note 6),
CR = torque motor input (amps) at 0.707 times resonant frequency (Note 7),
θ = active-end transducer output at 1.414 times resonant frequency (Note 6), and
CR = torque motor input (amps) at 1.414 times resonant frequency (Note 7).
θ = active-end rotation at 0.707 times resonant frequency (Note 7),
CR = torque motor input (amps) at 0.707 times resonant frequency (Note 8),
θ = active-end transducer output at 1.414 times resonant frequency (Note 7), and
CR = torque motor input (amps) at 1.414 times resonant frequency (Note 8).
NOTE 7—θ and θ will be functions of frequency for velocity and acceleration measuring transducers (see 8.1).
1 2
NOTE 8—If a current-measuring instrument is used, the units will be amperes. Alternatively, voltage drop across a fixed resistance may also be measured
and the units will then be volts.
By use of C and C , the torque motor rating, TMR, is obtained from:
1 2
TMR 5 0.5k C 1 C (12)
~ !
1 2
where:
k = apparatus spring constant, k (or for apparatus without springs, the calibrating rod spring constant, k ).
a rod
The torque applied to the top platen by the torque generator is given by:
τ 5 TMR·Trdg (13)
appl
τ 5 TMR·T (13)
appl rdg
where:
Trdg = input amps to the torque motor
T = input amps to the torque motor
rdg
TMR = torque motor rating from Eq 12.
8.5 Passive End Inertia and Torque Transducer Calibrations (only needed for Device Type 2):
8.5.1 A torque transducer generally consists of a metallic case containing a “spring” instrumented to measure strain where the
strain is proportional to the applied torque. The torque is applied to the spring through a sensing head protruding from the
transducer case. The sensing head must be rigidly connected to the passive end platen and provide the basis for the passive end
rotational inertia:
J 5 J 1J (14)
p passive platen sens head
D4015 − 21
where:
J = calculated using Eq 6-8, and
passive platen
J = frequently is provided by the transducer manufacturer.
sens head
8.5.2 Alternative methods provided in A2.3.
8.5.3 The torque transducer sensitivity is given by the manufacturer and must be traceable to a government standards agency. The
torque measured by the torque transducer is calculated from:
TTrdg
τ 5 (15)
TT
TTsen
TT
rdg
τ 5 (15)
TT
TT
sens
where:
TT- = Torque Transducer sensitivity typically in units of mV/(N-m)
sens
TTrdg = Voltage reading (mV) for the torque transducer.
TT = Torque Transducer sensitivity typically in units of mV/(N-m)
sens
TT = Voltage reading (mV) for the torque transducer.
rdg
9. Procedure
9.1 Test Setup—The exact procedure to be followed during test setup will depend on the apparatus and electronic equipment used
and on methods used for application, measurement, and control of the static axial and lateral stresses. However, the specimen shall
be placed in the apparatus by procedures that will minimize the disturbance of the specimen. Particular care must be exercised
when attaching the end platens to the specimen and when attaching the vibration excitation device to the platens. A temporary
support as discussed earlier may be needed. For cases where isotropic static stresses are to be applied to a membrane-enclosed
specimen, liquid- or air-confining media may be used for dry or partially saturated specimens. For tests where complete saturation
is important, a liquid-confining medium shouldshall be used. Where the vibration excitation device is located within the pressure
chamber, an air-liquid interface is acceptable as long as if the liquid covers the entire membrane that encloses the specimen.
9.2 Electronic Equipment—The Turn off the power supplied to the torque motor should be switched off. motor. Connect the torque
motor to the sine wave generator (with amplifier, if required). Connect the vibration transducers to the readout instruments.
Gradually apply power to the torque motor and adjust the readout instruments according to the instruction manuals for these
instruments.
9.3 Measurements:
9.3.1 Device Type 1:
9.3.1.1 Measurement of Resonant Frequency—The motion of the active-end platen in conjunction with the applied torque is used
to establish resonance. Resonance is defined as the lowest frequency where the torque is 90 degrees out-of-phase with the rotational
acceleration or displacement. This phase relationship can be detected by observing the Lissajous figure on an oscilloscope with
the torque input signal and rotational acceleration or displacement plotted as x-y. (Note 89) At the 90 degree phase relationship
the figure will be an ellipse with its axes vertical and horizontal. If a velocity transducer is used for rotational measurement, the
system resonance occurs when the Lissajous figure forms a straight, sloping line. It is recommended that the frequency be
measured with a digital electronic frequency meter and be recorded to at least three significant figures.
9.3.1.2 The determination of the lowest resonance resonant frequency can be done by setting the torque excitation frequency (for
example, 10 Hz) and power to as low a value as practical. Then increase the frequency of excitation until the system resonant
frequency is obtained.
NOTE 9—The phase relationship between two signals may also be computed by measurement of the time difference between zero crossings of the two
signals divided by the period of the oscillations (period = ) multiplied by 360 gives the phase in degrees. If the signals are not clean sine waves,
frequency
then a spectral analysis will have to be performed to get accurate values for magnitude and phase (or real and imaginary components) of the
rotation/torque ratio. The magnitude of the rotation/torque ratio multiplied by the cosine of the phase gives the real component of the rotation/torque ratio
D4015 − 21
and the same ratio multiplied by the sine of the phase gives the imaginary component of the rotation/torque ratio.
9.3.1.3 Measurement of Strain—The strain amplitude measurements shall be made only at the system resonant frequencies. Thus,
for a given torque, the vibration motion transducer outputs recorded at the system resonant frequency give sufficient information
to calculate strain amplitude. To increase or decrease strain amplitude, the applied torque must be increased or decreased. After
making a change in torque, the procedure of 9.3.1.1 must be followed to establish the corresponding system resonant frequency
before the rotation transducer output can be used to establish the new shear strain amplitude value.
9.3.1.4 Measurement of System Damping—Associated with each shear strain amplitude and system resonant frequency is a value
of damping. Two methods are available for measuring system damping: the steady-state vibration method and the amplitude decay
method. Both methods should give similar results. The steady-state method is easier and quicker. It is generally always used and
the amplitude decay method is used for occasional spot-checking. For the steady-state method, the active-end transducer output
and the applied torque must be measured at each resonant frequency. The calculations are outlined in the following section. For
the free-vibration method, with the system vibrating at the system resonant frequency, cut the power to the vibration excitation
device (see Note 9) and record the output of the rotation transducer used in establishing resonance as a function of time. The
shut-off mechanism must create an open circuit with the vibration excitation device and cannot be done by switching off the power
to amplifier. Without an open circuit, damping will be induced by current flow in the circuit. This gives the decay curve for free
vibration. The calculations for damping are outlined in the following section.
NOTE 9—The shut-off mechanism must create an open circuit with the vibration excitation device and cannot be done by switching off the power to
amplifier. Without an open circuit, damping will be induced by current flow in the circuit.
9.3.2 Device Type 2:
9.3.2.1 Measurement of Resonant Frequency—This is the lowest frequency at which the active end rotation is a maximum. In
addition to measuring the frequency, magnitude of motion and magnitude of torque, the phase between the motion at the active
end and the torque at the passive end must be determined (see Note 89).
9.3.2.2 Measurement of Strain Amplitude—The strain amplitude measurements shall be made only at the system resonant
frequencies. Thus, for a given torque, the vibration motion transducer outputs recorded at the resonant frequency give sufficient
information to calculate strain amplitude. To increase or decrease strain amplitude, the applied torque must be increased or
decreased. After making a change in torque, the procedure of 9.3.2.1 must be followed to establish the corresponding resonant
frequency before the rotation transducer output can be used to establish the new shear strain amplitude value.
9.3.2.3 Measurement of System Damping—Damping is determined from steady-state measurements of torque measured at the base
of the specimen (passive end), amplitude of motion of the active end and the phase difference between them as described in the
next section.
10. Calculation
10.1 General—Calculations require the apparatus calibration factors and the physical dimensions and mass of the specimen at the
time resonant measurements are made. In addition, for each static axial and lateral stress condition, one data set shouldshall be
measured for each vibration strain amplitude. A data set consists of: duration of vibration (this time can be used to calculate the
number of vibration cycles), system resonant frequency, active-end transducer output for both type devices. For Device Type 1
additionally, the reading associated with the applied torque, and if the amplitude decay method of measuring damping is also going
to be used, the free-vibration amplitude decay curve. For Devi
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