Experiment-55 S
STRAIN GAUGE MEASUREMENT OF YOUNG’S MODULUS USING
CANTILEVER
Dr Jeethendra Kumar P K
KamalJeeth Instrumentation and Service Unit, RMV 2nd Stage, Bangalore 560 094, INDIA.
Email: jeeth_kjisu@rediffmail.coml Abstract
Young’s modulus and strain per unit mass at a distance x from the load are determined using a metal strain gauge in full bridge circuit and aluminum cantilever. The values obtained are verified with the standard values.
Introduction
A cantilever is a common experiment performed in laboratory using traveling microscope. The traveling microscope measures the depression caused by the acting force on the cantilever. The depression also can be measured using a strain gauge. This is the electronic method of determining young’s modulus.
Strain gauge
Figure-1 Strain gauges
A strain gauge is a specially designed resistor whose resistance varies by stretching and compressing it. The change in the resistance is of the order of 1% or less. Electronic circuits can detect such a small resistance change. A strain gauge [1] is made of a conducting wire arranged in a wave pattern as shown in Figure-1. The conducting wire is etched on metal clad plastic film.
On the plastic film, the active region or the body of the resistance is clearly visible for nacked eye. Around the active region boundary, marks are made to align strain gauge while fixing [2].
The strain gauge is designed to paste to a surface using glue. When there is a deformation in the surface over which the gauge is pasted the resistance of the gauge change. Hence, the deformation taking on the surface is directly sensed by the strain gauge. Normally the percentage change in the resistance is around twice the change in strain. This gain factor two is designated as GF. This is not a universal constant but vary with gauge.
Metal alloys such as constantan, nichrome, dynaloy, stabiloy and platinum wires are used to make strain gauges. The resistance of the conducting strain gauge wire is given by
L
R A
= ρ …1
Where R is the resistance of the conducting wire A is the cross sectional area of the wire L is the length of the wire
ρ is resistivity of the material of the wire.
Force acting on the surface of the gauge causes elongation or contraction in the length of the wire, which results in resistance variation. However, the change in the wire length is fraction of millimeter still it could cause change in the resistance. The fractional change in the resistance to fractional change in the length is called gauge factor and designated as GF.
LR RL L
/ L
R / GF R
∆
= ∆
∆
=∆ …2
Semiconductor strain gauges are also available which has large gauge factor and more sensitive.
Table-1 shows a comparison between metal strain gauge and semiconductor stain gauge [3].
Semiconductor strain gauge are more sensitive but less accurate, whereas metal strain gauge are less sensitive and more accurate. Therefore, for accurate measurements metal strain gauges are preferred.
Table-1
Parameters Metal Strain Gauge Semiconductor Stain gauge Range 0.1 to 40,000µε 0.001 to 3,000µε
GF 2 to 4.5 50 to 200
Resistance 120,350,600,…5K 1K to 5K Tolerance 0.1% to 0.2% 1% to 2%
Size (mm) 0.4 to 150 1 to 5
Comparison between a metal strain gauge and a semiconductor strain gauge
Strain gauges are used to measure strain, force, pressure and flow. The metallic strain gauges are prepared by photo etching technique. Semiconductor strain gauges are made using piezoresistive material. Materials such as Silicon or Germanium are used. Semiconductor strain gauges are very small with very high sensitivity. However, semiconductor strain gauges are more sensitive to temperature in comparison with metal strain gauge. Semiconductor strain gauges are used for
force measurements; hence, it is also referred as displacement sensor. Displacement is proportional to force.
Strain
Force Acting
Contraction in the width
Elongation in the length Width
Length
Figure-2 Force acting on a fixed object fixed at one end
Strain is the amount of deformation of a body due to the application of force. Strain is denoted as ε and defined as the fractional change in the length as shown in Figure-2. The strain produced is given by
L L
= ∆
ε …3
Where ε is the strain
L is the length of material
∆L is the increase in the length Comparing equation 3 with 2 we can write
GF R / R
=∆
ε …4
The length of the object is increased and width is decreased due to the force. The increase in the length is positive strain or tensile, the decrease in the width is negative strain or compression.
The strain gauge if pasted on to the surface can sense the elongation and contraction taking in the material.
The elongation and compression is very small of the order 1% or less. To detect such small change Wheastone’s bridge is used as shown in Figure-3. The bridge will be balanced when
3 4 2 1
R R R
R = …5
The output voltage of the bridge is given by
ex 2 1 4 3
o 3 V
R R
2 R R
R
V R
− +
= + …6
Where Vex is excitation voltage
Vex
Vo R1
R2 R3
R4
Figure-3 Wheastone’s bridge
Any one arm of the Wheastone’s bridge can be replaced by a strain gauge for example R1 and R4
as shown in Figure-3 such a bridge is called half-wave bridge. To increase the sensitivity of the bridge all the four arms are replaced by strain gauge as shown in Figure-4. Two strain gauges A and B are used to detect the elongation or positive strain and two more strain gauges C and D are used to detect the contraction or the negative strain. Such a four arm active bridge is called full- wave bridge.
Vex
Vo RG-dR
RG-dR RG+dR
RG+dR DA C
B
C D
A B
A and B to detect elongation
C and D is to detect contraction Force
(a) (b)
Figure-4 (a) Full-wave bridge (b) The strain gauges are glued to a metal cantilever
The output voltage of the full-wave bridge is given by [4]
VO = - GF ε Vex …7
The equation is much simplified by activating all the arms of the bridge. The excitation voltage Vex is generally a dc voltage. A low voltage is preferred to avoid heating effect. In this experiment Vex is taken as 1.25volts. This selection guarantees very low temperature effect on the strain gauge.
Equation –7 shows that the strain produced is directly proportional to the output voltage of the full-wave bridge.
F ex
O G V
− V
=
ε …8
The output voltage is of the order of few microvolts, which need to be amplified to milli volt level in order to detect using digital milli voltmeter. An instrumentation amplifier is used to amplify the signal [5], which also convert the double ended-input in to single ended output as shown in Figure-6.
A RG+dR Vex
RG+dR
RG-dR Vo B
C RG-dR
D PT1
10K
Figure-5, Balancing the Bridge
An Opamp based instrumentation amplifier is used for this experiment. The amplifier detects only the difference mode signal rejecting the common mode signal. The gain of the amplifier is set to 10 by selecting RF =100K and RI = 10K. A zero setting for the bridge and offset zeroing is used in the circuit. A 200mV DPM is used to detect the output with its decimal fixed at the second digit. The DPM reads output voltage in milli volts from which strain is calculated using equation-8.
A RG+dR 10K D
RG-dR B
C RG-dR PT1 Vo
RG+dR Vex
15.00
200mV DPM +12V
-12V
0.1 100K
100K 10K
10K 10K
1 5 TP2 20K
741
+ -
3 2
6
74
Figure-6, Strain amplifier circuit
Figure-6 shows complete circuit diagram of strain amplifier. TP2 is offset adjustment of the Opamp, which is used as zero setting when there is no load on the strain gauge.
Stress
A cantilever is a mechanical arrangement in which one end of a rectangular scale or strip is fixed rigidly and the other end is free. The common example of a cantilever is the diving springboard in the swimming pool, which is fixed at one end. The force acting on the free end elongate upper surface and contract lower surface. The strain gauges are pasted to both the surfaces to detect the elongation and contraction. If F is the force acting on the cantilever then,
x 3 F Ybt
2ε
= …9
Where Y is the young’s modulus of the material of the scale b is its breadth
t is its thickness
x is the distance between the force acting point to the center of the strain gauge If m is the weight hanging at the free end of the cantilever, then force exerted is mg. substituting for F in equation 9 we get
ε
= bt2 mgx
Y 3 …10
X Strain Gauge
Force Scale
Fixed at one end
Figure-7, Measurement of distance x
In equation –10, 3g/bt2 is constant for a given cantilever hence the young’s modulus Y depends only on m/ε. By determining mass per unit strain (m/ε), Young’s modulus is determined.
Apparatus Used
An aluminum cantilever of 9 inch long and 1.5 inch wide is fitted with full bridge strain gauge as shown in Figure-8, a digital strain voltage indicator and 100x5 gm slotted weight.
Figure-8 Cantilever fitted with full bridge strain gauge
Fixing of strain gauge to the aluminum scale
The aluminum scale is polished at its center using zero number shine paper . The center of the scale is marked with a pencil. The strain gauge is pasted lengthwise using nitrous-based glue.
Two strain gauges pasted with its center coinciding with the scale center on each side of the scale as shown in Figure-8. Figure-9 shows detailed procedure of pasting strain gauge to the aluminum scale. The strain voltage indicator circuit is shown in Figure-6, which measures the strain voltage.
Top
Bottom A
B
C
D
9 inch 1.5 Inch
Center Line
Slot to fix weight
Figure-9, Four strain gauges are fixed to the aluminum scale
The complete experimental setup is shown in Figure-10, and same is available with KamalJeeth Instrumentation & Service Unit.
Figure-10, Young’s Modulus Experimental Setup
Experimental Procedure
1. The dimensions of the cantilever are noted by measuring its breadth and thickness up to 0.01cm accuracy using vernier calipers and screw gauge.
b = 3.85cm t = 0.20cm
2. The cantilever is fixed to its stand and screwed tightly. The gauge factor GF and exitation voltage is noted from the label printed on the stain voltage indicator.
GF = 2.1 ± 2% Vex = 1.25 Volts
The strain gauge is connected to the strain voltage indicator using D type connector.
3. The strain voltage indicator is adjusted to zero using zero-set control knobs (fine-offset and coarse-bridge balance) of the indicator. By watching the stability of zero- setting for 1 or 2 minutes a 100 gm weight hanger is fixed to the cantilever a distance x from the center line marked on the cantilever. The distance x is noted.
x = 10.3cm
4. The steady reading in the strain voltage indicator is noted in Table-2. Strain is calculated from eqauation-8.
F ex
O G V
= V
ε 30.47x10 6
25 . 1 x 1 . 2
mV 08 .
0 −
=
=
Table-2
Distance X=10.3Cms
Strain gauge output voltage (mV) εεεε x 10-6 m/εεεε x 106 Weight (gms) Trial-1 Trial-2 Trial-3 Average
0 0 0 0 0 - -
100 -0.08 -0.08 -0.08 -0.0800 30.47 3.28
200 -0.18 -0.17 -0.17 -0.1733 66.01 3.02
300 -0.27 -0.26 -0.26 -0.2633 100.30 2.99
400 -0.36 -0.35 -0.35 -0.3533 134.59 2.97
500 -0.45 -0.43 -0.43 -0.4366 166.32 3.00
600 -0.50 0.051 -0.51 -0.5066 192.99 3.10
Average m/εεεε 3.06 x 106
Distance X=9.0 Cms
Strain gauge output voltage (mV) εεεε x 10-6 m/εεεε x 106 Weight (gms) Trial-1 Trial-2 Trial-3 Average
0 0 0 0 0 - -
100 -0.05 -0.06 -0.05 -0.0533 20.30 4.926
200 -0.13 -0.14 -0.13 -0.1333 50.78 3.938
300 -0.20 -0.21 -0.21 -0.2066 78.70 3.811
400 -0.27 -0.28 -0.28 -0.2766 105.37 3.796
500 -0.34 -0.35 -0.35 -0.3466 132.03 3.787
600 -0.42 0.043 -0.42 -0.4233 161.25 3.721
Average m/εεεε 3.81 x 106 Strain Voltage at different loads
5. Trial is repeated by increasing the weight in steps of 100gms up to a maximum of 500 or 600 Gms. The corresponding readings are tabulated in Table-2. The average value of m/ε is calculated and Y is calculated.
6. To increase the accuracy of measurement, the weight hanger is removed and the strain indicator is monitored few minutes to see its zero setting. Wait until the zero setting is OK.
Restart loading weight at the same position with and another set of readings are taken.
Likewise, three trials are taken for each position.
ε
= bt2 mgx Y 3
11 )
6 6.45x10 10
x 28 . 3 04 ( . 0 x 85 . 3
3 . 10 x 980 x
3 =
= Dynes/cm2
7. Experiment is repeated by hanging the weight at 9 cms from the strain gauge center. The corresponding values are tabulated in table-2 and average value of Y is calculated for aluminum cantilever.
Results
The results obtained are tabulated in Table-3.
Table-3
Distance “x” 10.3cm 9.0cm
Strain/ unit mass = 1/(m/ε) 0.326 µ 0.262µ Young’s modulus of Aluminum 6.45 x1011 6.54 x1011
Y average 6.495x1011 dynes/cm2
Standard value 7.07x1011 dynes/cm2
Young’s modulus of aluminum
Discussions
A new method of determining young’s modulus using strain gauge is introduced in this experiment. A series of experiments will be presented in future volumes of LE dealing with strain gauge. The electronic part is much simpler using instrumentation amplifier. Students can do this experiment easily using locally available quick fix glue for fixing the strain gauge. The young’s modulus obtained in this measurement agrees well with standard value; 8% tolerance is observed.
The method can be used for brass, MS, copper cantilevers also.
References
[1] Robert F Coughlin and Frederick F Driscoll, Operational amplifier and Linear Integrated Circuits, 3rd Edn, Prentice-Hall, 1987, Page 209.
[2] Strain gauge laboratory, http:/www.personal.Dundee.ac.uk/~gathomso/strain.htm [3] Watt Kester, Editor, System Application Guide, Section-1,6, analog Devices Inc, 1993.
[4] http:/zone.in.com/devzone/conc.
[5] R F Coughlin and F F Driscoll, Operational Amplifier and linear Integrated circuits, 3rd Edn,1987, PHI, Page-209.
