In the glossary to the Pressure
Reference Section, “strain” is defined                                               Fv

as the ratio of the change in length to                              1
the initial unstressed reference                                 3
length. A strain gage is the element                                     L

that senses this change and converts
it into an electrical signal. This can be                        4
                                                                                                                          3                      4
accomplished because a strain gage
changes resistance as it is stretched,                                                                           1                                                    h
or compressed, similar to wire. For                Figure C - Bending Strain
example, when wire is stretched, its
cross-sectional area decreases;                                                                                                                                   b
therefore, its resistance increases.
The important factors that must be                       3                   4                              Figure E - Shear Strain
considered before selecting a strain
gage are the direction, type, and
resolution of the strain you wish to                                                                                               4
                                               b                                                                                             3
measure.                                                                                  FA
To measure minute strains, the user                                                                                                                                   Z

must be able to measure minute                       h                                                               45                              Z
resistance changes. The Wheatstone                           1                   2

Bridge configuration, shown in Figure               Figure D - Axial Strain
                                                                                                       MT                 2            1
B, is capable of measuring these                                                                                                                              Y
small resistance changes. Note the                                                                                                     L

signs associated with each gage                                                                             Figure F - Torsional Strain
numbered 1 through 4. The total
strain is always the sum of the four
strains.                                    would be 4 times the strain on one                         sectional modulus is (bh2/6).
                                            gage. See Figure C.                                        Strain gages used in the bending
                                                                                                       strain configuration can be used
                                                                 If total strain is four               to determine vertical load (F );
                                                                 times the strain on                   this is more commonly referred to
                                                                 one gage, this                        as a bending beam load cell.
                                    +      –                     means that the
       VIN                                                       output will be four        F = E B (Z)/ l = E B (bh 2⁄6)/ l
   REGULATED                                                     times larger.
        DC                                                       Therefore, greater      2) AXIAL STRAIN equals axial
                                                                 sensitivity and            stress divided by Young’s
                                    –       +
                                 2             3                 resolution are             Modulus.
                                                                 possible when                 EA = oA /E      oA = FA /A
                                                                 more than one
                                                                 strain gage is used.       Where axial stress (oA) equals
                                      VOUT                                                  the axial load divided by the
  Fig. B                                                         The following              cross-sectional area. The cross-
  Wheatstone Bridge                                              equations show the         sectional area for rectangles
                                                                 relationships              equals (b x d). Therefore, strain
The total strain is represented by a      among stress, strain, and force for               gages used in axial
change in V . If each gage had the
             OUT                          bending, axial, shear, and torsional              configurations can be used to
same positive strain, the total would     strain.                                           determine axial loads (F (axial)).
be zero and V would remain
                                          1) BENDING STRAIN or moment
unchanged. Bending, axial, and                                                                     F (axial) = E A bh
                                                strain is equal to bending stress
shear strain are the most common                divided by Young’s Modulus of            3) SHEAR STRAIN equals shear
types of strain measured. The actual            Elasticity.                                 stress divided by modulus of
arrangement of your strain gages will                                                       shear stress.
determine the type of strain you can          B = oB/E      oB = MB/Z = F (l )/Z
measure and the output voltage                                                                     = /G         = F x Q/bI
change. See Figures C through F.                Moment stress (oB) equals
                                                bending moment (F x l ) divided             Where shear stress ( ) equals
For example, if a positive (tensile)            by sectional modulus. Sectional             (Q), the moment of area about
strain is applied to gages 1 and 3,             modulus (Z) is a property of the            the neutral axis multiplied by the
and a negative (compressive) strain             cross-sectional configuration of the        vertical load (F ) divided by the
to gages 2 and 4, the total strain              specimen. For rectangles only, the          thickness (b) and the moment of


   inertia ( I ). Both the moment of                 where torsional stress ( ) equals      a gage factor of 2.0, Poisson’s Ratio
   area (Q) and the moment of                        torque (Mt) multiplied by the          of 0.3, and it disregards the lead wire
   inertia ( I ) are functions of the                distance from the center of the        resistance.
   specimen’s cross-sectional                        section to the outer fiber (d/2),
   geometry.                                                                                This chart is quite useful in
                                                     divided by (J), the polar moment       determining the meter sensitivity
         For rectangles only                         of inertia. The polar moment of        required to read strain values.
        Q = bh 2⁄8 and I = bh 3⁄12                   inertia is a function of the cross-
                                                     sectional area. For solid circular     Temperature compensation is
   The shear strain ( ) is                           shafts only, J = (d)4⁄32. The          achieved in many of the above
   determined by measuring the                       modulus of shear strain (G) has        configurations. Temperature
   strain at a 45° angle, as shown in                been defined in the preceding          compensation means that the gage’s
   Figure E.                                         discussion on shear stress. Strain     thermal expansion coefficient does
                                                     gages can be used to determine         not have to match the specimen’s
                 = 2 X @ 45°                         torsional moments as shown in          thermal expansion coefficient;
   The modulus of shear strain (G) =                 the equation below. This               therefore, any OMEGA® strain gage,
   E/2 (1 + ). Therefore, strain                     represents the principle behind        regardless of its temperature
   gages used in a shear strain                      every torque sensor.                   characteristics, can be used with any
   configuration can be used to                                                             specimen material. Quarter bridges
   determine vertical loads (F ); this                       Mt = (J) (2/d)                 can have temperature compensation
   is more commonly referred to as                              = G (J) (2/d)               if a dummy gage is used. A dummy
   a shear beam load cell.                                      = G ( d 3⁄16)               gage is a strain gage used in place
                                                                                            of a fixed resistor. Temperature
             F = G ( ) bI/Q                                   Ø = MTL/G(J)                  compensation is achieved when this
                 = G ( ) b (bh ⁄12)/(bh ⁄8)
                                 3       2                                                  dummy gage is mounted on a piece
                                                                                            of material similar to the specimen
                 = G ( )bh(2/3)                                                             which undergoes the same
                                                                                            temperature changes as does the
4) TORSIONAL STRAIN equals                                                                  specimen, but which is not exposed
   torsional stress ( ) divided by
   torsional modulus of elasticity (G).
   See Figure F.
                                                  T  he following table shows how
                                                  bridge configuration affects output,
                                                                                            to the same strain. Strain
                                                                                            temperature compensation is not the
                                                                                            same as load (stress) temperature
            = 2 x @ 45° = /G                      temperature compensation, and             compensation, because Young's
                                                  compensation of superimposed              Modulus of Elasticity varies with
                 = Mt(d/2)/J                      strains. This table was created using     temperature.

                                  POSITION           SENSITIVITY     OUTPUT PER

                                                                                                                                      STRAIN GAGES
                   BRIDGE         OF GAGES           mV/V @             @ 10 V             TEMP.    SUPERIMPOSED
 STRAIN            TYPE           FIG. C-F           1000            EXCITATION            COMP.    STRAIN COMPENSATED
                    ⁄4            1                  0.5             5     V/              No           None
 BENDING           1
                    ⁄2            1, 2               1.0             10     V/             Yes          Axial
                   Full           All                2.0             20     V/             Yes          Axial
                    ⁄4            1                  0.5             5     V/              No           None                          E
                    ⁄2            1, 2               0.65            6.5    V/             Yes          None
    AXIAL          1
                    ⁄2            1, 3               1.0             10     V/             No           Bending
                   Full           All                1.3             13     V/             Yes          Bending
                    ⁄2            1, 2               1.0             10 V/                 Yes          Axial and Bending
                                                                     @ 45°F
TORSIONAL          Full           All                2.0             20 V/                 Yes          Axial and Bending
                                                                     @ 45°F
Note: Shear and torsional strain = 2 x   @ 45°.

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