strain measurement 1

8/2/2019 Strain Measurement 1

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Department of Mechanical & Manufacturing Engineering Faculty of Engineering, University of Putra Malaysia

1

F

L

L +∆L

A0

F

Fig. 1 Material bar in tension

Fig. 2 Bonded metallic strain gauge

C a r r i e r

Active grid length

S o

l d e r t a b s

STRAIN MEASUREMENT 1

(BENDING AND TORSION)

Introduction

Strain is the amount of deformation of a material due to an applied force or stress, and measured by astrain gauge that consists of a pattern of resistive foil which is mounted on a backing material. As thematerial is deformed, causing its electrical resistance to change. This resistance change is related to thestrain by the quantity known as gauge factor. Therefore, one of the most widely used methods of experimental stress analysis is based on the strain gauge.

This manual contains some fundamental theory for understanding the experiment, description of theapparatus and experimental procedure for measurement of the bending and torsion strains.

Objective

The objectives of this experiment are1. To determine the strain in bending and torsion for various load configurations.

2. To develop an understanding of electrical strain gauges apparatus, and to verify its accuracy.

Theory

1. Strain

When a material is stretched due to the external forces and the length of the object increases from L toL +∆L (Figure 1), the ratio ∆L /L is called strain ε . The deformation is related to the forces F [N] by Hooke’s law and the elastic modulus E [N/m2 or Pa] in the expression

0// EAF L L ε =∆= (1)

As the ratio of deformation is often very small, it is oftenrepresented in a unit of 10 – 6 or µstrain.

2. Strain Gauge

A strain gauge (Figure 2) can be used to measure the strain of amaterial. The most common type of strain gauge consists of aflexible backing which supports a metallic foil pattern etchedonto the backing material. As the material is deformed, the foilpattern is deformed (Figure 3), causing its electrical resistanceto change. This resistance change, usually measured using aWheatstone bridge circuit (Figure 4), can be used to calculate theexact amount of deformation by means of the quantity knownas the gauge factor .

The gauge factor of a strain gauge relates strain to change inelectrical resistance, which is the ratio between fractional

change of resistance and strain. The gauge factor GF is definedby the formula

ε

R R

L L

R R G GG

F

/

/

/ ∆=

∆

∆= (2)

where R G is the resistance of the undeformed gauge, ∆R is the change in resistance caused by strain. Formany electrical resistance strain gauges, the gauge factor is 2.1 and the resistance about 120 Ω.




2

Fig. 3 Visualization of the working concept behind the straingauge on a material under exaggerated bending

Terminals

Higher resistance

Lowerresistance

Strainsensitive pattern

Area narrows,Resistance increases

Area thickens,Resistance decreases

Tension: C

R 1

R 2 R 3

R 4

V O V EX +

+ – –

Fig. 4 Wheatstone bridge circuit

R 2 R 3

R 4

V O

V EX

+

+ –

R 1+∆R

Fig. 5 Quarter-bridge circuit

–

3. Strain Gauge Measurement

The Wheatstone bridge is said to transducer strain into a voltage. The Wheatstone bridge is simply a set of fixed and/or variable resistances R i, arranged in a diamond pattern as shown in Figure 4. The excitation voltage V EX , is one output to the bridge, whose output is V O.

The output from the Wheatstone bridge can be expressed as:

) )( ( 4321

4231

43

4

21

1

EX

O

R R R R

R R R R

R R

R

R R

R

V

V

++

−=

+

−

+

= (3)

The bridge is balanced and produced no output ( V O = 0).

43

4

21

1

R R

R

R R

R

+=

+or

43

3

21

2

R R

R

R R

R

+=

+(4)

In order to calculate the output from a bridge, assume that R 1 is anactive strain gauge and it has changes by ∆R due to strain. Then, theoutput voltage is,

EX

4321

4231O

) )( (

)( V

R R R R R

R R R R R V ⋅

++∆+

−∆+= (5)

If R 1 = R 2 = R 3 = R 4

EX

22

O )2 )( 2(

V R R R

R R R R V ⋅

∆+

−∆+= (6)

Since R may be regarded extremely larger than ∆R ,

EX FEX O4

1

4V ε GV

R

R V ⋅⋅=⋅

∆= (7)

This is the output for what is termed a Quarter -Bridge (Figure 5).

Ideally, the resistance of the strain gauge changes only in response toapplied strain. However, strain gauge material, as well as the

specimen material to which the gauge is applied, will also respond tochanges in temperature. Strain gauge manufacturers attempt tominimize sensitivity to temperature by processing the gauge materialto compensate for the thermal expansion of the specimen materialfor which the gauge is intended. While compensated gauges reducethe thermal sensitivity, they do not totally remove it.

By using two strain gauges in the bridge, the effect of temperature can be further minimized. Forexample, Figure 6 illustrates a strain gauge configuration where one gauge is active ( R +∆R ), and a secondgauge is placed transverse to the applied strain. Therefore, the strain has little effect on the second gauge,

Active gauge ( R +∆R )

Dummy gauge ( R , inactive)

Specimen

F F

Fig. 6 Use of dummy gauge toeliminate temperature effects




3

R 2 R 3

R 4

V O

V EX

+

+ –

R 1+∆R

R 2 – ∆R – + –

R 3+∆R

R 4 – ∆R

Fig. 8 Full-bridge circuit

R 2 R 3

R 4

V OV EX

+

+ –

R 1+∆R

R 2 – ∆R

– –

called the dummy gauge. However, any changes in temperature will affect both gauges in the same way.Because the temperature changes are identical in the two gauges, the ratio of their resistance does notchange, the voltage V 0 does not change, and the effects of the temperature change are minimized.

The sensitivity of the bridge tostrain can be doubled by making both gauges active in a Half-

Bridge configuration.For example, Figure 7 illustratesa bending beam application withone bridge mounted in tension( R 1+∆R ) and the other mountedin compression ( R 2 – ∆R ).

This half-bridge configuration, whose circuit diagram is alsoillustrated in Figure 7, yields an output voltage that is linear andapproximately doubles the output of the quarter-bridge circuit.

Finally, the sensitivity of the circuit can be further increased by making all four of the arms of the bridge active strain gauges in aFull-Bridge configuration. The full-bridge circuit is shown inFigure 8.

4. Bending and Torsion Strains Measurement

Bending (Figure 9)

Using the theory of bending at the point of attachment of thestrain gauge

R E y σ I M /// == (8)

where M = Bending moment [Nm]I = Second moment of area [m4 ]= bd 3/12 ( b : width, d : thickness)

σ = Surface stress [N/m2 ] y = Half of the thickness of cantilever (= d /2) [m] E = Modulus of elasticity [N/m2 or Pa]R = Radius of curvature of cantilever [m]

At the middle plane of the cantilever, the length is unchanged by bending. Hence the increase in length of AB (Figure 10) due tobending can be written as

θ d y θ d R θ d y R CD ABdL ⋅=⋅−⋅+=−= )( (9)

Strain is defined as change in length per unit length, that is

R y θ d R θ d y L dL ε / )/( )( / =⋅⋅== (10)

From the theory of bending, the bending strain is

EI My R y ε // == (11)

EX FEX O V ε GV R

R V ⋅⋅=⋅

∆=

Fig. 7 Half-bridge circuit

EX FEX O2

1

2V ε GV

R

R V ⋅⋅=⋅

∆=

F Gauge in tension( R 1+∆R )

Gauge in compression( R 2 – ∆R )

L

W

b

d

Fig. 9 Bending cantilever beam with quarter-bridge

Fig. 10 Beam subjected to purebending, after the moment M has been applied

M M

y y

R

d θ A B

C D




4

Torsion (Figure 11)

The comparable theoretical equation is used for the torsionspecimen

L θ Gr τ J T /// == (12)

where T = Torque [Nm]

J = Polar moment of inertia of tube [m4 ]= π ( d 04 – d 14 )/32

( d 0: outer diameter, d 1: inner diameter)τ = Surface shear stress [N/m2 ]r = Outside radius of tube [m]G = Modulus of rigidity [N/m2 or Pa]θ = Angle of twist [rad]

The shear stress τ acts circumferentially and has to be accompanied by a system of complementary stresses including diagonal tensile and compressive stresses which are perpendicular to each other.Hence, there are equal direct strains along opposing 45º helices on the surfaces of the tube given by

EJ Tr Eτ ε // == (13)

and the meter will indicate 2ε

.

Description of Strain Gauge Apparatus

The HSM18 Electrical Resistance Strain Gauge Apparatus(Figure 12) has been designed to illustrate the basic featuresof electrical resistance strain gauges and their application tomeasurement of strain and the derivation of stress levels, inbending, torsion, tension and compression.

The apparatus is a read-out strain meter in a base box to which a pillar carrying an aluminium alloy cantilever has beenfixed. An electrical resistance strain gauge has been fixed tothe top surface of the cantilever 150 mm from the loading

point. A temperature compensation gauge is supplied fixedto a small piece of aluminium alloy. The basic circuit of the Wheatstone bridge is laid out on the top of the base,showing the use of a zeroing control. An analogue meter with a centre zero scale has been designed to read true strainin units of micro-strain, the calibration being achieved by adjusting the gain of the meter amplifier during assembly.

The torsion accessory consists of an aluminium alloy tube with loading arm welded across one end. Aclamp is provided to enable the tube to replace the cantilever strip used above, the loading arm being sethorizontally. A load hanger can be suspended on the vertical axis of the tube, or at horizontaleccentricities of 50 or 100 mm. The torsion bar having two gauges bonded orthogonally at 45º.

Specimen and Equipments

1. HSM18 Electrical Resistance Strain Gauge Apparatus2. Load hanger 0.5 N with C hook 3. Weights: 5 N, 10 N4. Stirrup5. Allen key and spanner6. Vernier caliper7. Bending and torsion specimens (E = 69000 N/mm2 )

Fig. 12 HSM18 Electrical ResistanceStrain Gauge Apparatus

T

r

Fig. 11 Torsion specimen with half-bridge




5

Procedures

Bending Strain Measurement

1. Measure the width, thickness and length of the specimen.2. Switch on the apparatus and adjust the set zero apex potentiometer to zero the meter. Note whether

any drift of the zeroed reading occurs as the strain gauges warm up. To show the effect of temperature warm the temperature compensation gauge by placing one’s finger on it.

3. Suspend the C hook of load hanger in the groove at the end of the cantilever. This will upset the zeroreading which should now be re-zeroed.

4. Press downward on the end of the cantilever and observe the direction in which the meter reads. Liftthe end of the cantilever and note that the meter reads in the reverse direction. Hence the polarity of the reading determines whether the gauge is in tension or compression.

5. Load the cantilever to 30 N by 5 N increments and read the meter at each increment.6. Repeat the readings as the load is removed. Enter the result in the following Table 1.

Torsion Strain Measurement

1. Measure the inner and outer diameters of the specimen.2. Place the torsion specimen and clamp.3. Connect the two pairs of leads from the torsion tube to the pairs of terminals, noting where gauge A

is connected.4. Switch on the apparatus and adjust the apex potentiometer to zero the meter.5. Re-zero if drift occurs as the gauges warm up.6. Place the load hanger at zero eccentricity and record the strain readings as the 30 N load is added by

10 N increments to the hanger.7. Note any meter reading, and check that the meter returns to zero when the loads are removed.8. Move the load hanger to 50 mm offset. Record the strain readings as the 30N load is added by 5N

increments to the hanger. Repeats the readings as the weights are removed.9. Repeat the step 8 for 100 mm offset. It will be necessary to hold the base box to prevent it being

toppled over the eccentric load.

Results

Bending Strain Measurement

1. Show all the measurements of specimen.i) Width b [mm]ii) Thickness d [mm]

2. Record the values of bending strain ε , and calculate the theory values of strain in Table 1.

Table 1 Experimental results of bending strain measurement

3. Tabulate the result and plot on the graph of indicated bending strain ε against load W .

4. Draw the best fit straight line through plotted points and add theoretical line calculated from the sametable.

Strain Reading ε [µε ]Load W [N]

Incrg. Load Decrg. Load Theory

0

5

1015

20

25

30




6

5. By using the gauge factor GF, calculate the change of resistance as a percentage of the nominal gaugeresistance ∆R /R (%) for the maximum strain indicated in the experiment.

Torsion Strain Measurement

1. Show all the measurements of specimen.i) Outer diameter d 0 [mm]

ii) Inner diameter d 1 [mm] 2. Record the values of torsion strain, and calculate the diagonal strain in Table 2.

Table 2 Experimental results of torsion strain measurement

3. Tabulate the result and plot on the graph of indicated torsion strain ε against load W .

4. Draw the best fit straight line through plotted points and add theoretical line calculated from the sametable.

5. Use the best fit straight line to determine the relationship between torsion and shear strain.

Discussion

1. Discuss on the obtained graphs.2. Compare the experimental results with the theory.3. How accurate would the both experiments have been in assessing the stress, where the strain gauges

were attached?4. If a strain gauge was use on an elastic materials, which was non-linear (i.e. did not obey Hooke’s law),

how could stress be determined?5. Discuss on the factors that can be affected to the experimental results.

Conclusion

1. Give an overall conclusion based on the obtained experimental results.2. Conclude on the applications of the experiment.

References

1. Hi–Tech. Instruction Manual for HSM18 Electrical Resistance Strain Gauge Apparatus – Measurement of Strain .Hi–Tech. 1998.

2. Arges. K.P. and Palmer. A.E. Mechanics of Materials . McGraw-Hill. 1963. 3 National Instruments. Measuring Strain with Strain Gauges . Available: http://www.ni.com, 2007.4. Kyowa. How Strain Gauges Work. Available: http://www.kyowa-ei.co.jp, 2007.5. Vishay. Force and Torque Measurement . Available: http://www.vishay.com, 2007.

Eccentricity [mm] Load W [N] Meter Reading ε [µε ] Diagonal Strainε [µε ]

0 10

20

30

50 / 100 5

10

15

20

25

30

strain measurement 1

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