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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                                                                     45°

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                                                                                                45°                    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 (Fn);
                                                                 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 n = E e (Z)/ l = E
   REGULATED                                                     times larger.             eB(bh2⁄6)/l B
        DC                                                       Therefore, greater      2) AXIAL STRAIN equals axial
                                    –       +                    sensitivity and            stress divided by Young’s
                                 2             3                 resolution are             Modulus.
                                                                 possible when
                                                                 more than one                 EA = oA /E      oA = FA /A
                                                                 strain gage is used.
                                      VOUT                                                  Where axial stress (oA) equals
  Fig. B                                                         The following              the axial load divided by the
  Wheatstone Bridge                                              equations show the         cross-sectional area. The cross-
                                                                 relationships              sectional area for rectangles
The total strain is represented by a      among stress, strain, and force for               equals (b x d). Therefore, strain
change in V . If each gage had the
             OUT                          bending, axial, shear, and torsional              gages used in axial
same positive strain, the total would     strain.                                           configurations can be used to
be zero and V would remain
                   OUT                                                                      determine axial loads (F (axial)).
unchanged. Bending, axial, and            1) BENDING STRAIN or moment
shear strain are the most common                strain is equal to bending stress                  F (axial) = E e A bh
types of strain measured. The actual            divided by Young’s Modulus of
                                                Elasticity.                              3) SHEAR STRAIN equals shear
arrangement of your strain gages will                                                       stress divided by modulus of
determine the type of strain you can        eB = oB/E oB = MB/Z = Fn(l )/Z                  shear stress.
measure and the output voltage
change. See Figures C through F.                Moment stress (oB) equals                          g = t/G            t = Fn x
                                                bending moment (Fn x l ) divided         Q/bI
For example, if a positive (tensile)            by sectional modulus. Sectional
strain is applied to gages 1 and 3,             modulus (Z) is a property of the            Where shear stress (t) equals
and a negative (compressive) strain             cross-sectional configuration of the        (Q), the moment of area about
to gages 2 and 4, the total strain              specimen. For rectangles only, the          the neutral axis multiplied by the

                                                                                             compensation of superimposed
   vertical load (Fn ) divided by the                    g = 2 x e@ 45° = t/G                strains. This table was created using
   thickness (b) and the moment of                            t = Mt(d/2)/J                  a gage factor of 2.0, Poisson’s Ratio
   inertia ( I ). Both the moment of                                                         of 0.3, and it disregards the lead wire
   area (Q) and the moment of                         where torsional stress (t) equals      resistance.
   inertia ( I ) are functions of the                 torque (Mt) multiplied by the
   specimen’s cross-sectional                         distance from the center of the        This chart is quite useful in
   geometry.                                          section to the outer fiber (d/2),      determining the meter sensitivity
                                                      divided by (J), the polar moment       required to read strain values.
         For rectangles only                          of inertia. The polar moment of
        Q = bh 2⁄8 and I = bh 3⁄12                                                           Temperature compensation is
                                                      inertia is a function of the cross-    achieved in many of the above
   The shear strain (g ) is                           sectional area. For solid circular     configurations. Temperature
   determined by measuring the                        shafts only, J = p (d)4⁄32. The        compensation means that the gage’s
   strain at a 45° angle, as shown in                 modulus of shear strain (G) has        thermal expansion coefficient does
   Figure E.                                          been defined in the preceding          not have to match the specimen’s
                                                      discussion on shear stress. Strain     thermal expansion coefficient;
              g=       2 X e@ 45°                     gages can be used to determine         therefore, any OMEGA® strain gage,
   The modulus of shear strain (G) =                  torsional moments as shown in          regardless of its temperature
   E/2 (1 + m ). Therefore, strain                    the equation below. This               characteristics, can be used with any
   gages used in a shear strain                       represents the principle behind        specimen material. Quarter bridges
   configuration can be used to                       every torque sensor.                   can have temperature compensation
   determine vertical loads (Fn ); this                                                      if a dummy gage is used. A dummy
                                                              Mt = t(J) (2/d)                gage is a strain gage used in place of
   is more commonly referred to as
   a shear beam load cell.                                       = g G (J) (2/d)             a fixed resistor. Temperature
                                                                 = g G (p d 3⁄16)            compensation is achieved when this
             Fn = G (g ) bI/Q                                                                dummy gage is mounted on a piece
                                                              Ø = MTL/G(J)                   of material similar to the specimen
                 = G (g ) b (bh3⁄12)/(bh2⁄8)                                                 which undergoes the same
                 = G (g )bh(2/3)                                                             temperature changes as does the
                                                                                             specimen, but which is not exposed
4) TORSIONAL STRAIN equals                                                                   to the same strain. Strain
   torsional stress (t) divided by
   torsional modulus of elasticity (G).
                                                   The following table affects output,
                                                   bridge configuration
                                                                        shows how            temperature compensation is not the
                                                                                             same as load (stress) temperature
   See Figure F.                                   temperature compensation, and             compensation, because Young's
                                                                                             Modulus of Elasticity varies with

    STRAIN         BRIDGE         POSITION            SENSITIVITY     OUTPUT PER            TEMP.     SUPERIMPOSED
                   TYPE           OF GAGES            MV/V @          m e @ 10 V            COMP.     STRAIN COMPENSATED
                                  Figs. C-F           1000 m e        EXCITATION
                    ⁄4            1                   0.5             5 m V/m   e           No            None
 BENDING           1
                    ⁄2            1, 2                1.0             10 m V/m      e       Yes           Axial
                   Full           All                 2.0             20 m V/m      e       Yes           Axial
                                                                                                                                       STRAIN GAGES
                    ⁄4            1                   0.5             5 m V/m   e           No            None
                    ⁄2            1, 2                0.65            6.5 m V/m     e       Yes           None
    AXIAL          1
                    ⁄2            1, 3                1.0             10 m V/m      e       No            Bending
                   Full           All                 1.3             13 m V/m      e       Yes           Bending
                    ⁄2            1, 2                1.0             10 m V/m      e       Yes           Axial and Bending
                                                                      @ 45°F                                                           E
TORSIONAL          Full           All                 2.0             20 m V/m      e       Yes           Axial and Bending
                                                                      @ 45°F

Note: Shear and torsional strain = 2 x   e @ 45°
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