NOTEBOOK
or, in terms of relative resistance:
Gage Factor
dR= [dL_ dA]+ dp
R
L
A
p
Fundamentals
The electrical resistance strain gage has its origins
in a series of experiments performed by Professor
William Thomson (Lord Kelvin) to investigate the
electrodynamic properties of metals. One of his
findings, reported to the Royal Society of London in
1856, was that the electrical resistance of certain
wires varied when stretched. 1 This phenomenon
has since been quantified; the ratio of the relative
resistance change to the relative length change is
the gage factor, G, of the strained conductor:
G= dR/R = dR/R
dL/ L
(l)
EL
where R is the initial resistance, Lis the initial
length, and ELis unit strain along the length.
The strain gage is largely in debt to these simple
relationships for its well established reputation as
the most practical deviee for measuring mechanical
strains. They are all one really needs to know and
understand to successfully use a strain gage. To
understand why the strain gage works so incredibly well requires a more substantial effort. Come
along! That story is a very interesting mixture of
engineering mechanics and atomic physics.
(4)
Now, if the conductor is a wire of circular cross
section:
(5)
where ris the initial radius of the wire. Differentiating once again:
(6)
dA= 27T rdr
Wllliam. Thomson,
Lord Kelvin ( 1824-
1907)
Scottish
mathematician,
physi
, inventor,
and
ch engtneer was a professor
of natural philosophy at Glasgow University for 53 years.
Lord Kelvin was
knighted for his theoretical and experimental work on
the Atlantic telegraph, as weil as his
many
er important
butions to both science and. engineering.
Substituting this into Eq. 4, and simplifYing:
Strain-induced resistance changes
(7)
The resistance of a conductor is a function of three
variables:
L
(2)
R=p -
The two terms within the brackets, of course, contain the two orthogonal unit strains, EL and E"' that
are produced when the wire is stretched:
A
dR
where p is the resistivity of the conductor material,
Lis the length of the conductor, and A is its crosssectional area. This expression can be differentiated to yield the sources of small changes in resistance:
p
pL
L
dR=-dL--dA+-dp
A
A2
A
(8)
p
Further, the radial strains are the compressive
Poisson strains corresponding to the longitudinal
tensile strain:
(3)
Page 2
dp
-=[E
-2E r ]+ R
L
(9)
January 1993
where vis the Poisson's ratio. By substituting this
result into Eq. 8 and simplitying, we obtain the
interesting result that the relative resistance
change would appear to be influenced by both the
mechanical and electrical properties of the strained
conductor:
dR= [EL(!+2v) ] +dp
-
R
•
(10)
p
We know from mechanics of materials that
Poisson's ratio of most metals varies from about
0.3 for strains in the elastic range to 0.5 for those
in the plastic range. That causes the term containing the Poisson's ratio in the preceding equation to
increase drastically when the conductor is
stretched past its elastic limit. And, that would
appear to cause the resistance to vary nonuniformly with strain. Fortunately, we shall soon see that
the resistivity term cornes to our rescue in this
regard.
Strain-induced resistivity changes
Our friends in materials science tell us that electrical currents are conducted through metals by
means of free electrons. With the aid of an elementary textbook on atomic physics 2 and a modest
investment in time to develop a rudimentary
understanding of the principles involved, one can
quickly accept that the resistivity of a conductor
can be expressed as:
(11)
where mis the mass of an electron, v0 is the average magnitude of the velocity of the electrons in
their random motion between ions within the conductor, n is the number of conduction electrons
available per unit volume, e is the electron charge,
and A is the average distance traveled by an electron between collisions with the ions within the
conductor.
•
Our needs are better suited if this expression is
rewritten as:
(12)
January 1993
where the electrons per unit volume, n, is replaced
by the total number of conduction electrons, N 0 , in
a conductor having a length, L, and a cross-sectional area, A. Now, recognizing that the electron
mass and charge are constants, we can differentiate this expression, divide through by the resistivity, and simplify to obtain the change in relative
resistivity:
(13)
Substituting this into Eq. 4, we obtain:
dR = dL + dv 0
2
R
L
v0
_
dA _ dN0
A No
(14)
This is a particularly nice result. The effects of the
area change produced by the Poisson strains cancel each other in the geometrie and resistivity
terms. As a result, the Poisson's ratio of the conductor can undergo its characteristic change in
value during the transition between elastic and
plastic deformation without affecting the rate at
which the resistance of the conductor changes
when stretched. With all mechanical properties
eliminated from further consideration, all that
remains to affect resistance is the relative change
in length of the conductor (unit strain) and, at the
atomic level, any strain-induced changes in the
number of free electrons, their velocity, and the
distance between their collisions with ions in the
conductor.
Strain-induced gage-factor changes
By substituting the preceding expression for relative resistance change into Eq. 1 and simplitying,
we obtain a broader expression for gage factor:
dRIR
dR/R
G=--=--=2+
dL/ L
EL
(dv 0 1vo -dAIA-dN,JNo)
(15)
EL
This result is less than ideal because gage factor is
clearly a function of strain. While the first term has
a constant value of 2 for all strains in all materials,
the value of the second depends upon the ratio of
strain-induced changes in the atomic structure of
the material to the strain producing them. Under
these circumstances, gage factor can be a constant
Page 3
NOTEBOOK
only when the sum of these changes is either zero
or is directly proportional to the strain producing
them:
·•
x rel
110
100
(dv 0 lv 0
-
dÀI À -dN0 1 N 0 ) =
(16)
kEL
!10
Unfortunately, that is seldom the case; less-thanideal gage factor behavior is the norm for most conductors.
80
70
Strain-induced resistance changes that occur at
the atomic level are influenced not only by composition of a material, but also somewhat by its thermomechanical history. The gage factor of a material that has been annealed may be quite different
from that of the same material after it has been
cold-worked, for example. The understanding of
just what happens to the atomic structure is a very
complicated subject of considerable interest in the
physics community. Indeed, it may be possible to
use this knowledge to design a metal alloy that
would have a constant gage factor for all strains.
The strain-gage pioneers, however, were a much
more pragmatic lot; they just stretched wires of
readily available metals and alloys, and measured
the resistance changes to establish the gage-factor
characteristics of those materials.
.tl!!:
1
•
/
(
..----
f
c 1
0
•
ARI1
4
1
•
•
Ille l in•
0
1
15
1
·2
v
·4
·•
0
311
llO
1
/R
&
10
1
&
4
STRAIN
1
•
0
IOXIO...
Fig. 1 - Resistance change and stress as afunction
of strainfor nickel (after heat treatment).
Page 4
10
50
40
50
10
10
z
STRAIN
li
Fig. 2- Resistance change and stress as afunction
of strain for nickel wire containing 1/4% iron.
Pure nickel was one of many unsatisfactory
materials investigated. 3 Its resistance initially
decreases with increasing tensile strain at a rate
corresponding to a gage factor of about -14. At still
higher strain levels, as shawn in Fig. 1, its resistance reaches a minimum before beginning to
increase at a rate equivalent to a gage factor of
approximately 2. If nickel is alloyed with a small
amount ofiron (Fig. 2). the resistance initially
increases until a similar dip in resistance is initiated at an elevated strain level. Things are clearly in
turmoil in the atomic structure; a change in the
number of free electrons, the distance between collisions, or both apparently causes the gage factor
to change within certain narrow bands of strain for
these materials.
A steel piano wire is better behaved. While its
resistance always increases with elastic strain, as
shawn in Fig. 3, the gage factor of this material
does vary significantly with strain magnitude. The
resistance changes for strains near zero are at a
rate equivalent to a gage factor of about 2. Over the
first few thousand microstrain, the resistance
January 1993
x 101
22 i 10-s
A gage factor for engineers
200
The gage factor, as we have presently defined it, is
a differentia! gage factor; it is the gage factor associated with an infinitesimal resistance change from
any initial value of resistance. The data in the preceding figures suggest that the resistance change
per unit of strain for a material can be quite different at different strain levels. Whenever this happens, the differentia! gage factor will have a different value at each of these strain levels. Because
each gage factor defines the ratio of relative resistance change to strain over a very small range, we
can correctly view each one of them as the local
gage factor at a specifie strain level.
180
180
140
120
1
~/R
8
/}_
100
lbs /in1
80
60
40
The engineer has little use for such limited gage
factors. While the strains measured by engineers
are (hopefully) small, any truly infinitesimal strains
are usually insignificant strains. Accordingly, an
20
0
•
28 x 10'"'
STRAIN
Fig. 3- Resistance change and stress as afunction
of strainfor steel {piano wire).
accelerates with strain until the changes correspond to a gage factor of about 3. 7 for the remaining strain levels in the linearly elastic range. Once
again, this behavior can be attributed to changes
at the atomic level.
•
J
24
20
AR /R
2
e
4
1
J
3
'
'
4
•
6
STRAIN
12
15
Fig. 4 - Resistance change as afunction of strain
during repeated cycles of loading beyond the yield
point for iron.
'
January 1993
J
li
11
{i
A
.\,
W
'
;;.~
e
Although the change in Poisson's ratio between
elastic and plastic strains has no direct effect on
gage factor, sorne materials do have very different
gage factors for strains above and below the yield
point. The gage factor of iron wire is similar to that
of steel piano wire in the elastic region. Yet when
strain levels exceed the yield point (Fig. 4) resistance begins changing at a rate equivalent to a
gage factor of about 2.0. Similar observations have
been made of resistance changes in many other
materials. This kind of behavior suggests that, during plastic deformation facilitated by the creation,
movement, and eventual annihilation of dislocations, the electronic variables reach a kind of
steady state. If this happens such that deformation
proceeds at a constant volume, then resistance
varies with strain in the plastic range at a rate corresponding to a gage factor of 2.0.
v
Page 5
NOTEBOOK
engineering gage factor must be based on small,
but finite, strains that produce small, but finite,
resistance changes:
= LlR/ R = i1RI R
GE
i1LIL
(17)
EL
The engineering gage factor for a small, but
finite, deformation may be quite different from the
differentiai gage factors at the various strain levels
within the bounds of the deformation. Consider
once more, for example, the strain-induced resistance changes of the steel piano wire. When relative resistance change is plotted as a function of
Lagrangian unit strain, the differentiai gage factor
at any strain level is nothing more than the slope of
the curve at that strain level (Fig. 5). Therefore, the
differentiai gage factor for the piano wire increases
from 2.0 at zero strain, to about 3. 7 at 6000JLE.
The engineering gage factor, based on the total resistance change, increases from 2.0 at zero strain
to 3.4 at 6000JLE.
x 10-1
An engineering gage factor for ali strains
The engineer charged with measuring engineering
strains would prefer that the engineering gage factor remain the same for all values of strain. That,
we shall soon see, never happens for real materials. In fact, even the ideal conductor does not have
a constant engineering gage factor.
Suppose for the moment that sorne alloy could
be formulated for which the number of free electrons, their velocity, and the distance between their
collisions with ions in the conductor were unaffected by strain. If this were possible, then gage factor
would be solely a function of strain-induced geometry change; we know from Eq. 15 that the differentiai gage factor at all strain levels would be
exactly 2.0 for this most ideal of strain gage alloys.
Unfortunately, the same is not true for the engineering gage factor.
While no changes occur in the electronic structure when our idealized material is deformed, its
resistivity is still dependent upon the volume of the
conductor:
22
p=
20
18
2mv 0 AL
N0 e2 A
= kAL = kV
(18)
where k is constant at all strain levels. The resistance of a conductor having such a resistivity can
now be written as:
16
14
L
L
A
A
2
R = p- = kAL- = kL
12
M/R
(19)
This expression can now be used to obtain the
resistance change caused by a finite change in
length of the conductor:
10
8
(20)
6
The relative resistance change is:
4
i1R
R
G=2.0
Fig. 5 - Differential gage factor at a point may differ
from engineering gage factor over a strain range.
Page 6
(21)
With the elimination of both the cross-sectional
area and the constant, k, the relative resistance
change is once again independent of all materials
properties. Whether the Poisson's ratio changes
January 1993
with strain is not relevant; whether elastic defor-
mation is accompanied by a volume change, or
plastic deformation occurs at constant volume, is
immaterial. The relative resistance change is a
function only of the fini te change in the relative
length of the conductor.
case of iron (Fig. 6), the differentiai gage factor
changes from about 4, to nearly 2. As this happens, the engineering gage factor also undergoes
an equally abrupt change. However, the effect on
its magnitude is less dramatic because of the averaging effect of the resistance changes preceding
yielding.
The engineering gage factor is obtained by
dividing the relative resistance change by the
Lagrangian engineering strain along the length of
the conductor:
x 10- 3
16·~~--------------------~----------,
(22)
12
•
This interesting result has many equally interesting implications. Foremost among them is the
observation that resistance changes arising from
finite changes in geometry cause the engineering
gage factor to increase with strain magnitude. For
the special case of our ideal material in which all
resistance changes are caused by changes in
geometry, the differentiai gage factorisa constant
of 2 at all strain levels while the engineering gage
factor increases from 2 at zero strain, to 2+EL at all
other strain levels.
The engineering gage factor for less-than-ideal
materials is similarly affected as well. The deviation
of differentiai gage factor from 2 in real materials
for elastic strains is, of course, primarily attributable to strain-induced electronic changes. In
the case of the iron, for example, these account for
nearly half the gage factor in the elastic range.
Since elastic strains are relatively small, the
increase in engineering gage factor due to geometrie changes can be readily overshadowed by
increases or decreases in the engineering gage factor due to electronic changes. Consequently, if the
differentiai gage factors of a material are larger
than 2.0 during elastic deformation, the engineering gage factor may actually decrease - rather
than increase - with increasing strain until well
after the onset of plastic deformation.
•
The variations in gage factor due to electronic
changes generally disappear for plastic strains. If a
material undergoes an abrupt transition from elastic tc plastic deformation, the differentiai gage factor will change quickly at the yield point. In the
January 1993
~
- 4
8
l---=--1'3.79
a::
<1
0
Engineering
GageFa7
ti
ca
u..
Q)
Cl
ca
(!l
4
2.06
~rential
- 2
Gage Factor
o._______...L--------L.----------'o
0
3
6
9 x 10-3
Strain
Fig. 6 - Differential and engineering gage factors
may vary with strain at different rates.
Many materials, like Nichrome (Fig. 7 next page).
for example, experience a graduai elastic-to-plastic
transition, and the portion of the gage factor associated with electronic changes varies more slowly
with strain. Simultaneously, the geometry-related
portion continues to increase with strain. When
this happens, the gage factor undergoes an even
less abrupt change at the onset of plastic deformation. Eventually, as constant-volume plastic deformation is attained, gage factor changes attributable
to geometry will overshadow any increase or
decrease caused by electronic changes. This is particularly true for very large strains of l% or more.
The quantification of gage factor variation with
strain in real materials is the subject of much
debate in the technicalliterature. While all the
Page 7
------
--
-
-~~
NOTEBOOK
t.R/R
4.0
<*----,2!:---4!------!:-8----;!8 X10"5 O
STRAIN
Fig. 7 - Resistance change, stress, and engineering
gage factor as ajunction ojstrainfor Nichrome
(soft).
geometrical considerations are rather straightforward, the electronic behavior of strained materials
is mu ch more difficult - if not impossible - to
comprehend. All that can be said with any degree
of certainty is that the engineering gage factor will
vary with strain to sorne extent. How much depends primarily upon the changes in electronic
structure at the lower strain in the elastic range;
for large strains in the plastic range, the geometrie
effects tend to dominate the changes in gage factor.
sensing circuit. And, insensitivity to temperature is
desirable because the sensing circuit cannat distinguish between temperature- and strain-induced
resistance changes in a gage. The resistance
behavior of Cupron, Capel, Advance, Constantan
and other similar alloys is nearly perfect for strain
gages because they have these traits. Containing
approximately 55% copper and 45% nickel, these
alloys can be formed into small wires and thin
foils. Originally developed as materials for constructing resistive elements, they are characterized
by a relatively high resistivity and relatively low
thermal coefficient of resistance.
The differentiai gage factor of fully annealed constantan wire (Fig. 8) is initially about 1.9 at zero
strain, increases to approximately 2.0 at its yield
point of about 2000J.LE, and continues to increase
with strains in the plastic region. On the other
hand, constantan previously work-hardened by
plastic deformation (Fig. 9) has a differentiai gage
factor of about 2.1 for all strains in its extended
elastic limit. In fact, any changes in gage factor in
this elastic region are so small that they are
13 :10"
1/
12
1
Il
0
9
1
R/R
7
In addition to a strain-insensitive gage factor,
strain gages need high resistances to minimize selfheating when they are installed in a resistance
Page 8
1
1
j
B
5
li->-
STVU ~
40
1
ll!Jc/R
1
6
Since our ideal material does not actually exist,
real strain gages must be made of less-than-ideal
conductors. Because the differentiai gage factor of
most conductors does tend toward a value of 2 for
strains in the plastic range, those with gage factors
of about 2 in the elastic range as well are the conductors that have the most uniform gage factor at
all strain levels. We are fortunate to have several
such materiais at our disposai.
60
lbs/ ;.,0
1
1 1
20
: 1_ 1
2
j j
10
Il/ 1
v
1
2
3
4
STRAIN
5
6
7X
Fig. 8- Resistance change and stress as ajunction
of strainjor constantan (soft).
January 1993
13
)( lo'
on·•
;1
v
12
Il
/
10
STRES/
8
8
/
7
llR/R
6
f------
5
/
1
1
v
110
100
/
J
/
120
90
;1
80
/
lbe/
/
50
~IR
4
40
/
2v v
3/
30
20
1/
•
is that the gage factor of gages is generally somewhat different than the gage factor of the conducting material of their grids.
130
y
1
10
3
2
STRAIN
4
5
)( Il>·•
Fig. 9- Resistance change and stress as ajunction
of strainfor constantan (hard).
generally beyond our abilities to accurately measure them. As such, they are of no practical significance, and the engineering gage factor ofworkhardened constantan can be considered to be constant for all strains in its elastic range of approximately 400ÜJ.LE. And, that makes constantan an
ideal material for general-purpose strain gages
designed and manufactured to measure the kinds
of strains most commonly encountered in loadbearing parts, members, and structures.
Gage factors for strain gages
•
The preceding discussion of gage factors has been
limited to unbonded conductors under uniaxial
tensile loads. They deform in a very straightforward
manner: when stretched, an attendant contraction
of their cross sections proceeds in accordance with
the inherent mechanical properties of the concluetor material. The sensing grids of strain gages,
however, are usually bonded conductors which
may be exposed to both tensile and compressive
force . transmitted to them from the surface to
which they are attached. The result, we shall see,
January 1993
Suppose that a constantan conductor is mechanically attached to the surface of a constantan
specimen with an electrically insulating adhesive.
Further suppose that an external force is subsequently applied to the specimen parallel to the conductor. The specimen is obliged to deform in direct
response to the applied load, but the conductor is
not. Rather, it must deform in response to the
equilibrium forces developed between surface and
conductor. These forces are transmitted from that
surface, through the adhesive, to the conductor.
Despite our best efforts, the conductor will experience a somewhat smaller deformation than the
specimen because of not only the localized reinforcement of the specimen by adhesive and wire,
but also the incomplete transmission of the surface
strain through the adhesive. Since it is the strain
in the unreinforced specimen that is of interest, the
gage factor of a bonded conductor normally will be
somewhat different than that of an identical,
unbonded conductor when specimen and unbonded conductor are exposed to equivalent strains.
Now suppose that the constantan conductor
could be bonded to the surface of the specimen in
such a way that no gage reinforcement or loss of
strain transmission occurs when loaded. Were this
possible, the longitudinal and transverse strains
would be identical in both the specimen and conductor. If a single force is applied to the specimen
parallel to the length of the conductor under these
conditions, the cross-sectional area of both conductor and specimen would contract due to the
transverse Poisson strains produced by elongation.
If the specimen were subsequently reloaded with
forces both parallel and transverse to the concluetor in such a way that the same change in length of
conductor is produced, the change in cross section
would be either more or less than that produced by
the Poisson effect alone. Consequently, the volume
and cross-sectional area of the same conductor
would be different for identical changes in length
produced by different loading conditions. These differences would have little, if any, effect on resistance changes due to geometry because, as previously noted, changes in cross section are largely
t\ \
~
Page 9
NOTEBOOK
self-canceling. The same cannat be said for resistance changes due to alterations in the electronic
structure. These - arising from variations in the
number of free electrons, their speed, and distance
between collisions - are so sensitive to volume
change that they tend to cease altogether for plastic deformation proceeding at constant volume.
Accordingly, for a bonded conductor with differentiai gage factors of other than 2.0, the total resistance change in the conductor is also somewhat
sensitive to the transverse strains in the surface to
which it is bonded.
Strain gages are seldom made of a single length
of conductor. They more commonly contain a grid
of closely spaced, parallel conductors connected in
series by end loops. If these end loops are in the
plane of the gage, then sorne portion of each one
responds primarily to whatever transverse strains
might be present. That means the total resistance
change in the gage for a given change in gage
length is different whenever the transverse strains
are different. And, when the resistance change is
different for the same length change, that means a
different gage factor for each combination of longitudinal and transverse strains.
The manufacturer's gage factor
The foregoing suggests that a large number of variables play a role in determining the gage factor of a
strain gage. The conductor material itself is the
single largest variable in determining gage factor.
The gage manufacturer's influence is limited to the
design of the gage grid, selection and processing of
the grid material, and application of an insulating
backing that retains the grid shape during installation. The end user must complete the gage manufacturing process by selecting and applying a strain
gage adhesive; the extent of strain transmission
from the surface, through the adhesive, to the grid,
also hasan effect. And, finally, the state of stress
and strain in the application itself has an unavoidable influence on gage factor due to variations in
gage reinforcement, strain magnitudes, and the
ratio of transverse and longitudinal strains. While
gage factor can be determined for every possible
circumstance, such an undertaking is bath
impractical and unnecessary. Rather, a manufacturer's gage factor is measured to provide a hench
Page 10
mark indication of how a specifie gage res ponds to
a single, standard set of circumstances. Then,
when measurements must be made under other
conditions, adjustment can be made to account for
deviations from these standard conditions.
Similar procedures for determining manufacturer's gage factor are outlined in International
Recommendation No. 62 issued by the
Organisation Internationale de Métrologie Légale
(OIML), and in MethodE 251 from the American
Society for Testing and Materials (ASTM). Bath
require that testing fixtures be used to produce a
uniform uniaxial stress field corresponding to nominal principal strains of 0 ±1000fLE. The corresponding transverse strains are also set by a
requirement that the Poisson's ratio of the test
specimen be 0.28 ±0.01. The effects of gage reinforcement are minimized by utilizing relatively
thick steel test fb~:tures. Bath tensile and compressive engineering gage factors of strain gages bonded to these fixtures are determined from the measured difference in installed resistance of these
gages at zero strain and at full tensile and compressive load, respectively. The manufacturer's
gage factor is then reported as the statistical average of a number of these tensile and compressive
engineering gage factors measured under these
specifie test conditions. The manufacturer's gage
factor of most constantan foil gages is typically
between 2.05 and 2.10, depending primarily upon
the thermomechanical his tory of the grid material
and the grid design.
Once a strain gage is bonded to a surface with
modern adhesives for gage factor determination, it
is not easily removed. Accordingly, most strain
gages are sold as uncalibrated measurement
deviees. The manufacturer's gage factor supplied
with them is a statistical entity based on measurements actually made on other gages constructed
from the same lots of materials at the same time.
Because of unavoidable variations between gages,
sorne variation in gage factor is inevitable. Typically
equivalent to ±0.5%, or less, of the average gage
factor, this variation is reported as two standard
deviations; the gage factors of 95% of all untested
gages should fall within these limits of uncertainty.
January 1993
Axial gage factors and transverse sensitivity
By substitution of this expression, and simplification, the manufacturer's gage factor is:
The relative resistance change in a strain gage can
be expressed as the sum of resistance changes due
to uniaxiallongitudinal and transverse strains:
jjR
jjRL
jjRT
-=--+-R
R
R
(29)
(23)
These individual resistance changes can also be
expressed in terms of axial gage factors in the longitudinal and transverse directions:
While the Poisson's ratio of the test fixture, the
manufacturer's gage factor, and the transverse sensitivity coefficient are all known values, the axial
gage factor in the longitudinal direction generally is
not. Fortunately, it can be calculated through a
simple rearrangement of Eq. 29:
(30)
By substitution of this expression into Eq. 27, and
simplification, the engineering gage factor can be
obtained for any longitudinal strain in any strain
field:
The ratio of transverse to longitudinal gage factors
is the coefficient of transverse sensitivity for the
gage:
(25)
•
This coefficient is also determined by the gage
manufacturer in accordance with the previously
mentioned OIML and ASTM standards. A special
fixture is utilized to produce a uniaxial strain field.
The ratio of the resistance changes in gages
mounted parallel and transverse to this strain
yields the value of the transverse sensitivity coefficient for a particular combination of grid material
and design. These ratios are typically within the
range of ±0.02 for modern foil gages, and are usually reported as a percentage.
l+(ET/EL)K1 )
1- V 0 K 1
(31)
Transverse Sensitivity in Strain Gages.
Local gage factors for small strains
(26)
Suppose an installed strain gage has a grid for
which the ratio of the incrementai change in resistance, L1R, produced by an incrementai change in
its length, L1L, is constant:
jjR = k
jjL
Dividing by the longitudinal strain yields the engineering gage factor of the gage in any strain field:
(27)
(32)
Under these conditions, the engineering gage factor
is a constant for all strains producing changes in
the initial resistance, R 0 , and length, L0 :
jjRf Ro = k La =Go
jjLf L 0
R0
The manufacturer's gage factor is nothing more
than the engineering gage factor determined in a
uniaxial stress field on a material with a very specifie Poisson's ratio, V 0 , such that:
(33)
If that same strain gage is somehow stretched to
(28)
January 1993
(
Only wh en the ratio of the transverse and longitudinal strains corresponds to -v0 will the engineering and manufacturer's gage factor be equivalent.
Consequently, measurements made in other strain
fields, but based on the manufacturer's gage factor,
will require corrections for maximum accuracy.
These procedures are outlined in Measurements
Group Tech Note TN-509 entitled, Errors Due to
With Kt in hand, the total resistance change can
now be expressed in terms of the longitudinal gage
factor:
•
GE =GM
sorne other beginning length, L 1 , the corresponding
resistance, R 1 at that new length can be calculated
î<'~\
Page 11
NOTEBOOK
where L1R is the change in resistance from R 0 to R 1
corresponding to the strain, E 1 , produced in the
gage by the change in its length from L 0 to L 1 . In
general, the local gage factor will decrease with
increasing initial resistance.
121.00
'iii'
E
121.75
..c::
~
8s:::
.;
'iii
The manufacturer's gage factor is calculated on
the basis of the nominal resistance (RJ supplied
with the strain gages. Correction for the local gage
factor should be considered if the resistance of the
gage is substantially altered (from R 0 to R 1) by permanent deformation during handling, by installing
the gage at no load on a surface with a small
radius of curvature, or by preloading the gaged
specimen to high strain levels before initiating
strain measurements.
121.50
Q)
a:
Q)
Cl
al
(!)
121.25
121.00
1.001
1.000
1.002
Length
1.003
1.004
Gage factor variation with temperature
Fig. 10 - Local gage factor varies with both strain
magnitude and initial value of resistance.
Gage factor varies with temperature in a repeatable, but virtually unpredictable, manner. If the
general expression for gage factor (Eq. 15) is differentiated with respect to temperature, gage factor
variation with temperature is seen to be independent of the geometrie effect; the derivative of 2 with
respect to temperature is zero.
as follows:
(34)
Suppose that the gage at this new length (Fig. 10
for example) is now obliged to measure a small
strain that produces a corresponding incrementai
change in resistance and length. The gage factor
under these conditions is:
d( dv 0 1V 0
This new local gage factor is different, but the ratio,
k. of the incrementai resistance change to the incrementallength change is the same in both cases.
The two gage factors can be related through the k
that is common to both:
o Lo
1
~
Therefore, if a strain gage with a known gage factor, G0 , at sorne initial resistance, R 0 , is used to
measure a strain beginning at sorne other initial
resistance, R 1 , the new local gage factor, G 1 , at that
new initial resistance can be expressed as follows:
J
Page 12
(38)
The gage factor of constantan tends to increase
linearly with temperature at the rate of about 0.5%
per 100°F, while that of Karma decreases at a similar rate. Iso-Elastic alloy has a gage factor that increases with temperature in the lower range, peaks
at about 150°F, and then decreases with temperature above that. The manufacturer's gage factor at
temperature, Gy. can be easily determined from:
(36)
G, =G0 R0 .!::J_=G ( l+E, l=G0 (R 0 +llR/G0
R, L0
° 1 + G0 E1 j
R0 + ilR
dt
Ràther, how gage factor varies with temperature
depends upon how the strain-induced variations in
number of free electrons, their speed, and distance
between collisions are influenced by temperature.
Quantifying these effects by other than experimental methods is practically impossible.
(35)
k=G Ro =G !!i_
-d~~À-dN0 1N0 J
dG
dt
(39)
(3 7 )
,.
W
'
'
January 1993
where LlGM is the percentage variation from the
manufacturer's gage factor, GM, at + 75°F. These
data, also determined in accordance with methods
specified in the OIML and ASTM standards, are
also supplied by Micro-Measurements and other
gage manufacturers for selected grid alloys.
A summary of practical considerations
"Everything that happens, happens as it should,
and ifyou observe carefully, you willfind this to be
so."
- Marcus Aurelius Antoninus ( 121-180)
•
•
We are fortunate that a conductor should undergo
a change in resistance when stretched. How it happens at the atomic level is not easily observed, predicted, or understood. But, with everything happening as it should, these changes are sufficiently
repeatable that the relative change in resistance
can be related to the strain producing it through a
useful transfer coefficient known as the gage factor.
Gage factors of all real materials vary with strain
and temperature to sorne extent. This undesirable
trait has been minimized to acceptable levels in
commercially available strain gages through selection of conductor material. Work-hardened constantan is perhaps the best general-purpose strain
gage alloy. Gages made with this alloy have a gage
factor that is constant, for all practical purposes, at
about 2.05 to 2.10 within their elastic range of
strains, and that is very similar in the plastic range
as well. In the early days of strain gages, high gage
factors were desirable because of instrumentation
limitations. However, with the stable, high-gain
amplifiers that are available today, gage factors of
2.0 are more than sufficient for most applications.
Accordingly, grid materials other than constantan
are normally specified for special-purpose applications. Karma, a similar alloy having a slightly higher gage factor and much better corrosion resistance, is used in gages for making strain measurements at higher temperatures. Iso- Elastic alloy,
often the material of choice for gages requiring
fatigue resistance, has a quite nonlinear gage factor that is about 3.2 at +2000JLE and decreases
with strain. And, gages having grids of platinum
alloyed with 8% tungsten are sometimes used for
applications having low strain levels or requiring
limited Wheatstone bridge power; these strain
January 1993
TRANSVERSE SENSITIVITY, K 1 IN%
Fig. 11 - Transverse sensitivity errors for various
ratios of axial and transverse strains.
gages have a gage factor of about 4. 7 at +2000JLE
that decreases even more quickly with strain.
The manufacturer's gage factor can be used
without modification in many - if not most stress analysis applications. However, corrections
for the following should be considered when warranted:
Transverse sensitivity - Errors due to the transverse sensitivity of gages should be corrected if the
ratio of transverse-to-longitudinal strains is far
removed from 0.285, the Poisson's ratio at which
the manufacturer's gage factor was measured. This
condition is almost always present when three-element rosettes are engaged to determine the magnitude and direction of the maximum and minimum
principal strains. The range of errors is shown in
Fig. 11.
Large strains - Gages, particularly those with
gage factors far removed from 2.0, strained in
excess of their elastic limit, are vulnerable to
{\> \
~
Page 13
NOTEBOOK
significant strain-induced changes in gage factor.
Unfortunately, the extent of these changes is very
difficult to quantity by any method. A general
review of the technical literature on the subject
may be helpful, but for strains well into the plastic
region of deformation, a value somewhere between
2 and 2+E is likely to be a reasonable educated
guess.
Independent errors due to thermal output are produced simultaneously when the gage undergoes
temperature changes. These have more potential
for serious consequences because, unlike the percentage errors caused by gage factor variation with
temperature, they are absolute errors independent
of strain magnitude.
And, with that behind us, we are left only with
something called the instrument gage factor. But
that's another story for another day ... perhaps.
Gage reinforcement- Unencapsulated strain
gages with polyimide backings have a modulus of
elasticity of about 500 000 psi; the modulus is
about twice that for glass-fiber-reinforced backings. Accordingly, all gages produce sorne degree of
local reinforcement that effectively lowers the gage
factor. The effect is most pronounced on plastics
and other low-modulus materials. However, it can
usually be ignored without serious consequence if
the gages are installed on test specimens of rn etals
and other high-modulus materials, particularly
when the specimen thickness is ten or more times
that of the strain gage.
~
<~f!asuring Gage
,'f,, ~:,
,k,,
~'
' "
Determination of the gage factor of a bonded electrical resistance strain gage is a relatively inexpensive and simple student exercise involving a singleelement strain gage installed on a prismatic bar
subjected to cantilever flexure. Here's how.
Local gage factor - This correction is usually
unnecessary unless the installed resistance of the
gage varies significantly from the nominal resistance as supplied by the manufacturer. If not corrected for local gage factor, the measurements will
contain errors that, on a fractional basis, are
slightly less than the relative difference in nominal
and installed resistances.
The mechanics
The famous flexure formula for calculating the longitudinal stress, a 5 , produced on the surface of a
long, slender bearn by bending, is one of the most
widely used in all of mechanics:
Gage factor at temperature - Producing errors of
about 0.5% per 100°F, the temperature effect on
gage factor can be ignored except for substantial
temperature excursions from room temperature.
Mc
as =1-
1Thomson, W. (Lord Kelvin). On the
Electrodynamic Qualities of Metals, Phil. Trans.
Roy. Soc. (London}, 146 (1856), 649-751.
2 Richards,
C.W. Engineering Materials Science.
(Monterey, CA: Brooks/Cole Publishing Company,
1961).
3 Kammer,
..
W
For our cantilevered bearn with a rectangular
cross section, the location of the neutral axis, the
bending moment and the moment of inertia can be
expressed in terms of the a pplied load and physical
dimensions:
.
Page 14
(l)
where M is the moment producing the bending, c is
the distance from the neutral axis of bending to the
surface, and 1 is the moment of inertia of the bar's
cross section. This expression is particulary useful
because it places no restrictions on any of the variables so long as the cross sections of the bearn remain plane upon deformation by bending.
References:
E.W., and T.E. Pardue. Electrical Resistance Changes of Fine Wires during Elastic and
Plastic Strains, Froc. SESA, 7 1 (1949), 7-20.
factor
. (4,· S'~~l'l' ptudeot EXerQae\.
1
January 1993