FUNDAMENTALS AND APPLICATION OF STRAIN GAGES by bjb17276

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									                                                                    FSG 1
  FUNDAMENTALS AND APPLICATIONS OF STRAIN GAGES
WHAT IS STRAIN?




                                               L




                                                   L


                                 
For a member under tensile stress:

      L = deformation caused by the tensile stress.
      L = original length
       = applied stress (force/area)

By definition:

      Strain = L / L = 

                  Load
      Stress =         =
                  Area


STRESS - STRAIN DIAGRAM OF A MATERIAL


                                                        x
                 STRESS


                                      LINEAR ELASTIC STRAIN LIMIT




                                             STRAIN
                                                  FSG 2
ELECTRICAL RESISTANCE OF A WIRE AFFECTED BY STRAIN

                  l
Recall       R
                  a

         P                                                                           P



                       A
                                                  L                         L

When the wire has a strain of L / L in the longitudinal direction, the
corresponding strains in the lateral directions are:

              r      L
                                   Where:  (or  ) = Poisson’s ratio
              r       L
                                                  r = radius of wire

Then the change in cross-sectional area is:

              a = r2 – ( (r - r)2 )= 2rr + (r)2  2rr

                                 L                   L
              a = -2r2           , a = r2 = -2a 
                                 L                    L

R due to L:


            R     R     R                                 L     L
      R    L    a                                L  2 a  
            l     a                                 a     a      a

     R L a            a       L                            R L      L 
                , since     2    ,                                2   
     R   L   a             a       L                             R   L      L   


                       R                  
                            R  1  2      
                                                      = the Gage Factor
                       L                  L
                            L                 L
                                                                 FSG 3
               R
                    R   lies between approximately 1.9 and 2.1
               L
                    L




ELECTRICAL WIRE STRAIN GAGES


1. Wire Gage




2. Etched Foil Gage




3. Semiconductor Gage

      Gage Factors as high as 100


4. Deposited Gage

      Gage is deposited on the tested material
      Gage can be better positioned
      No bonding problem


5. Weldable Strain Gages
    Very convenient – easy to apply
                                                 FSG 4
RELATIONSHIP BETWEEN CHANGE IN RESISTANCE AND STRAIN

            R
                (GF)  where: GF = Gage Factor and  = strain
            R

     Example: What is the change in R (resistance) in the Gage?
                                  400 psi




                                                 E = 4.0 106 psi
              Strain Gage
              G.F. = 2.05
              R = 120 




Solution:  = 400 / 4 x 106 = 1 x 10-4 R / R = (2.05)(1 x 10-4) = 2.05 x 10-
4



                   R = (120 ) (2.05 x 10-4) = 0.0246  (not very large)

How can we measure such a small change in resistance?


WHEATSTONE BRIDGE
                                     i1 + i2


                                         R1                      R2

                  +
                VO                                i1        i2
                  _

                                         R4                      R3

                                                       RG



                                               Galvanometer - very
                                               sensitive amp meter
                                                                                                  FSG 5

For a balanced bridge, the current through the galvanometer is zero and
the voltage across the galvanometer is also zero.

                    i1 R1 = i2 R2                                            (1)

                    i1 R4 = i2 R3                                            (2)

Divide (1) by (2)

            R1 R 2                                                                      R2
                                            or                          R1 = R4
            R4   R3                                                                     R3

For R1 = R2 = R3 = R4 = Ro (initially) and if R1 changes by R
                                             i1 +i2


                                             R0 + R                               R0

                     +                                           i1     i2
                    VO
                     _
                                                                                             iG
                                                      R0                       R0

                         i1 +i2
                                                                      RG


                                                             _        V +

For small R :
                         Vo R
            IG =                        (ignoring the higher order terms)
                    4R o (R o  R G )

                 V =
                           Vo R R G
                                                  =
                                                           GF          RG
                                                                                   Vo
                         4R o (R o  R G )                  4         (R o  R G )

                                                          
                                    GF  1               
                                  =                         VO
                                      4  1 R O           
                                          R               
                                              G           
                                                                       FSG 6
                               GF
For RG  Ro,      V =            Vo . This is known as a quarter bridge.
                                4

For a strain gage located at position R1 in a Wheatstone bridge circuit,

                                   GF
                           V =        Vo
                                    4

In the previous example, if Vo = 20 volts,

            V =
                   2.05
                     4
                           
                        1 x10  4 20 V      = 1.025 x 10-3 volts


QUARTER BRIDGE WITH TEMPERATURE COMPENSATION


                               Active Gage     R1               R2

                    +
                   VO
                    _

                                               R4              R3


                        Dummy Gage
                        for Temperature                  V
                        Compensation



Since the active gage and the dummy gage are identical, the changes in R
in the two gages due to thermal effects are the same.

                           (R1)thermal = (R4)thermal


                                      R1 R 2
For a balanced bridge:                  
                                      R4 R3
                                                                        FSG 7
With the thermal effects on both gages, the bridge is still balanced.

                        R 1 + R 1  R   R
                                    1  2
                        R 4 + R 4  R4 R3

The circuit is self-compensating for thermal effects.


ALTERNATE CONNECTION FOR DUMMY GAGE (QUARTER BRIDGE)

                                                        Dummy Gage
                                                        for Temperature
                 Active Gage    R1                R2    Compensation


         +
        VO
         _

                                R4               R3



                                         V


                                              R1  R1 R 2  R 2
        R1 + R` = R2 + R2                           
                                                 R4        R3




SELF - TEMPERATURE - COMPENSATING GAGES

Gages with temperature sensitivity the same as the material it is placed on
- used with specified test materials.

Example: steel or aluminum compensated gages
                                                                FSG 8


USE OF TWO ACTIVE GAGES FOR INCREASED SENSITIVITY (more
voltage output for a given strain – higher output to noise ratio)
                                        




                               1        3




                                        


                     Gage 1
                                   R1             R2

          +
         VO
          _
                                                       Gage 3
                                   R4             R3



                                            V


                              R1 = R2 = R3 = R4

For a strain of :
                                    (GF) 
              V due to Gage 1 =           Vo
                                      4

                                    (GF) 
              V due to Gage 3 =           Vo
                                      4
                                                                   FSG 9
                                   (GF) 
         Total measured V =              Vo    but
                                     2


    General Strain Gage Equation:

                 (GF) 
         V =           N Vo
                   4

         Where N = sensitivity factor (or # of active gages) (N = 2 in this
         example).


HALF BRIDGE TO MEASURE BENDING STRAINS


                                Strain Gage 1



                                Strain Gage 2


                1 = strain at Gage 1 (tension)
                2 = strain at Gage 2 (compression)

                               2 = -1

                Gage 1                                         Gage 2
                                 R1                   R2
                (Tension)                                   (Compression)
        +
       VO
        _

                                 R4                   R3



                                           V

                                  (GF)  1
         V due to Gage 1 =                Vo
                                    4
                                                                                  FSG 10
                                            (GF)  2                (GF)  1
           V due to Gage 2 =                       Vo       =             Vo
                                              4                       4
                  (GF)  1
           TOTAL V =      Vo                           (N = 2)
                    2
SHEAR STRAIN MEASUREMENT


                                  X
                                                


                                                         L


                                   

                        x
Shear Strain             
                         l


MOHR CIRCLE FOR STRAINS
                         

                          0 /2
                                                                            

      4                               1           
                                                                       0



               -0/2                                      Gage 1
                                                                            Gage 4




                       At location of Gage 4,  = 4 = o/2

                       At location of Gage 1,  = 1 = o/2
                                                                    FSG 11
BRIDGE CONNECTION FOR SHEAR STRAIN MEASUREMENT


                                R                 R
                Gage 1
         +
       VO
         _
                Gage 4
                                R                 R



                                         V



FOUR ARM (FULL) BRIDGE


                                                      Strain Gages 1 & 3
        Strain Gages 1 & 3



        Strain Gages 2 & 4



        Side View                                     Top View



                                R                 R
                Gage 1                                       Gage 2
         +
       VO
         _
                Gage 4                                     Gage 3
                                R                 R



                                         V

                         V = (GF)  V0 , N = 4
                                                                    FSG 12

Problems associated with attaching the connecting wires.
TWO WIRE (leadwires) CIRCUIT

For Quarter Bridge




             Power                        V
             Supply




Disadvantages:

     1. The resistance of the leadwires will add to the resistance of the
        gage.

     **2. The effects of temperature on the leadwires will affect the strain
         readings.

Solutions:

     1. Use as short as possible leadwires with large cross-sectional area
        to minimize resistance.

     2. Use dummy gage with lead wires the same as the active gage

                                                          Active Gage




       Power                         V
       Supply
                                                          Dummy Gage
                                                                  FSG 13
The temperature effects on the leadwires will be self-compensating if the
leadwires to both the active gage and dummy gage are identical (same
length and type).

      Best Solution - Use a three-wire circuit. No DUMMY gage required




          Power                           V
          Supply




Thermal effects will be minimized if leadwires are identical. Note the wire to
the voltmeter is not a leadwire and is not part of the circuit.




CALIBRATION OF BRIDGE CIRCUITS (to verify that what you are
reading on the bridge output is correct)


                                                       RS
                                                        Shunting Resistor

                           R0                  R0

    +
   VO
    _

                           R0                  R0



                                     V
                                                                   FSG 14
When the calibrating resistance, RS is shunted across Ro, the resulting
drop in resistance of that arm is:

                                                           2
                                      R oR S            Ro
                          R  R o                  
                                     Ro  RS           Ro  R S

                       R    Ro                  Ro
                                                      for RS  Ro
                       Ro  Ro  RS               RS

                             1  R       1 Ro
                      V           Vo       Vo
                             4  Ro       4 RS

The measured V is similar to that produced by a strain of S

                                     N
         V due to a strain, S, =
                                     4
                                       GF  SVo

      N                     1 R                       1   Ro
Set
      4
        GF  S Vo  VS  4 R o Vo      S 
                                                   N GF R S
                                S




When RS is shunted across Ro, the resulting V is the same as that caused
by a strain of S.

                                                      1    Ro
               Output of VS indicates strain of
                                                   N GF  R S
                                                                          FSG 15


INSTRUMENTATION FOR MEASUREMENT OF DYNAMIC STRAINS

                                                            RS


                        R0                       R0
          Active Gage
+
VO
_

                        R0                   R0


                                                          Amplifier   Output to
                                                                      Oscilloscope




MEASUREMENT OF LARGE STRAINS

     R
        as high as 10%
     R
                               R
     V 
            R Vo
                                   R
                                        Vo
                                             
                                                    GF  Vo
          4 R  2 R           
                             4 1 
                                    1 R 
                                         
                                                    
                                                  4 1 
                                                         GF 
                                                               
                                   2 R                    2   




END OF TEST MATERIAL NUMBER 1

								
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