Strain Gauge Lab and Young¶s Modulus Measurement Engineering Mechanics 2: 16232
Jeswin Mathew Mathew 200901475 Electrical and Mechanical Engineering
Contents Introduction««««««««««««««««««««««««««««««««««.2 The Strain Gauge Experiment «««««««««««««««««««««««««.....2 Theoretical Background «««««««««««««««««««««««««««2 Surface Preparation and Bonding of the Strain Gauge ««««««««««««««3 Results««««««««««««««««««««««««««««««««««.4 The Young¶s Modulus Experiment«««««««««««««««««««««««««5 Theoretical Background «««««««««««««««««««««««««««5 Experimental Procedure and Apparatus«««««««««««««««««««....7 Results and Analysis««««««««««««««««««««««««««««8 Conclusion«««««««««««««««««««««««««««««««««...«11 References«««««««««««««««««««««««««««««««««..«11
1. Page | 1
Introdu cti on
The laboratory was divided into two sessions: During the first session a strain gauge was bonded onto a solid beam with elastic properties by first cleansing the surface using surface preparation techniques[2] and then bonding the strain gauge on to the µcleansed spot¶ (The reasons for this will be discussed later) using a permanent adhesive. The second session entailed the measurement of the Young¶s Modulus solid of abeam by applying a load on both ends of the bar and measuring the increments in strain () and the central deflection () caused due to increments in the load. The Young¶s modulus was approximated from the gradient of the graph of the load against strain and central deflection. Both these experiments and the relevant theories that apply to them will be discussed throughout the report.
2. T h
Str
i
Ga uge
Exp erim en t
2.1 Theoretical Background
2.11 Int odu tion
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The primary component using in the manufacturing of a strain ga uge is the strain-sensitive alloy, used in the manufacture of the foil grid (Refer to Figure1), the foil grid safely is mounted on an encapsulation or backing material and solder dots renders a conductive surface for soldering lead connections. If a deformation was introduced into the solid beam, this will subsequently deform the alloy in the foil grid, causingits electrical impedance to increase when the bonded surface is in tension, and decrease if the surface is in compression as shown in figure 1, and is usually measured using a wheat stone bridge; this resistance is related to the strain by a quality factor known as the gauge factor [1]:
, [1]
Equation 1
Where dR is the change in resistance due to deformation, Ro is the resistance when unstressed. Foil Grid
Backing Material or encapsulation Decrease
Figure 1: Bonded Strain Gauge 2.12 Types of strain sensitive alloys
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Solder Tabs
There are three types of strain sensitive alloys [2]: the Constantan alloy, Isoelastic alloy and the Karma alloy, the Constantan alloy is the most widely used.
I. II. III. IV.
Constantan Alloy: This alloy possesses a very high strain sensitivity (Gauge Factor) It is characterised by a good fatigue life and can handle relatively high elongations. It is suitable for the measurement of very large strains (5%). Annealed Constantan is a grid material normally used; Constantan in this form is very ductile and can handle higher strains (20%). Isoelastic Alloy: Isoelastic alloy (D alloy) has a very high superior fatigue life and also a high gauge factor which makes it very suitable for dynamic strain measurements [2]. Karma Alloy: This alloy is characterised by good fatigue life and excellent stability and is the primary choice for static strain measurements for long periods of time, usually months [2].
2.2 Surface Preparation and Bonding of the Strain Gauge 2.21 Surface Preparation Prior to the bonding of the strain gauge to the object, the surface had to be prepared using certain techniques. The purpose of surface preparatio n was to obtain a chemically clean surface, a surface whose roughness and surface alkalinity was similar to that of the strain gauge. The surface was prepared through the five procedures outline below in order [2]:
y
Degreasing the surface using a solvent Surface Abrading Marking the gauge layout lines Surface Conditioning Neutralising Surface Degreasing: This was performed to remove contaminants, greases and soluble chemical residues; during the experiment Acetone was used for the degreasing process. Also, careful methods were used while cleaning and drying it to eliminate any further contamination. Surface Abrading and m k out the layout lines: The surface abrading procedure was perfor m using a silicon carbide paper and m conditioner solution, and was done to rem ve an loosely bonded adherents such as scale, rust coatings and also to develop a surface texture suitable for bonding as shown in figure 2. The surface was wetted using Figure 2: Surface Abrading the m conditioner and was cleansed by rubbing the paper against the surface. . After drying, the layout where the strain gauge was to be bonded was m k using a ball point pen with a pair of crossed, perpendicular reference lines. ¤
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Page | 3
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y
Surface Conditioning and Neutralising: After the layout lines where market the metal conditioner that was used before was used to clean the surface again but this time with gauze using a single stroke and finally a neutraliser was used to provide maximum alkalinity for the strain gauge adhesives. Again when drying the surface was wiped with one single stroke to avoid dragging back on the contaminants from the previous stroke.
2.22 Strain Gauge Bonding y
y
y
The first priority was given into proper handling of the gauge due to the delicacy of the foil grid; manual tools were used to withdraw the gauge from its envelope, and then it was placed on a glass slab which was also degreased using acetone to remove contaminants. A length of cellophane tape (About 10cm) was stuck carefully on top of the gauge, the ends of the tape was stuck to the glass slab. The tape was then peeled off very carefully; the main purpose of this was to temporarily bond the gauge to the tape so that it can be transported to the prepared surface for permanent bonding as shown in figure 3.
Figure 3: Strain Gauge removed from glass slab.
The strain gauge was then positioned on the prepared surface as shown in the figure, first, before applying a Catalyst on the bottom of the strain gauge, which facilitates the bonding. Once this was done a strong adhesive was applied on the surface as shown in the figure 4 and the tape holding the gauge was rolled onto the surface. y
Figure 4: Application of Adhesive
y
After a minute, the tape was removed of leaving the strain gauge bonded on the surface, thus completing the process. The final step was to solder the leads onto the solder tabs (See Figure 1) for electrical connection.
2.3 Results The strain gauge was tested for both compressive and tensile stress and the resistances were found to vary accordingly for the scenarios as expected; when the beam was unstressed, the resistance was observed to be roughly 121 which can be taken to be . It was also observed that there were only obscure differences in the electrical resistances when the beam was stressed.
3 . T he Measu rem en t of the Youngs Modulu s Page | 4
3.
r ic l B c
r u d
!
M
M
The Distance µy¶ f rom Neutral Axis; +y f or above the axis and ±y f or below
Radius of Curvature of the Neutral Axis, µR¶
Figure 5: Bending of Beams [3] 3.11 The Ela stic Beam Theory If a b eam portraying elastic properties, of symmetrical cross-section is subject to a bending moment M, then stresses will occur along the sur f ace and also the beam will bend into a simple arc as shown in f igure 5. It can be noted f rom the f igure that the upper f ibres of the beam are in tension due to the increase in their length and the bottom f ibres will be in compression due to the converse occurring. F or a beam subject to bending moments as seen in the f igure, the stress and the Y oung¶s modulus can be related using equation [4] 2: "
Equation 2
Where, E is the Y oung¶s modulus of the material in N/m2, is the stress and is given by the equation Force/ Area, its units are N/m 2,
is the strain and is given by Elongation/O riginal Length. Another relation that encapsulates, the Y oung¶s modulus, stress and strain is shown in equation 3:
Where M is the moment in Nm,
Page | 5
Equation 3
I is the second moment of inertia across the cross section of the beam R is the radius of curvature of the neutral layer of the beam due to the bending moment M, y is the distance f rom the neutral axis to any point on the thickness (cross section) of the material. 3.12 Neutral Axis and Radius of curvature Neutral axis is a line through the thickness cross-sectional area of the beam where the length stays the same during tensile or compressive stress[4], and also the and experienced by the beam ± caused by bending moments as shown in f igure 5 - increases linearly as y increases i.e. the maximum strain is observed at the top sur fa ce of the beam which has the greatest distance f rom the neutral axis. Th e strain increases linearly and is related to y by the radius of curvature (R) f or the neutral layer using the equation below [4]:
Equation 4
mula by using Also the stress at distance y f rom the neutral axis can also be calculated f rom the above f or the relationship portrayed in equation 2:
Equation 5
The notation beside y indicates that y is positive f or distances above the neutral axis and negative f or below the neutral axis as shown in f igure 5. Th e neutral axis f or a beam of unif or m cross-section passes right through the centroid of its cross-section. Th e radius of curvature is a quantity that is practically impossible to obtain an absolute value and theref ore can only be approximated; when the beam is unstressed, the radius of curvature is undef ined. 3.13 Shearing Force and Bending Moment The shearing f orce in a beam at any section is the f orce transverse tending to cause it to shear across the section; the shear f orce will remain unif or m on an unloaded part and will change abruptly at a concentrate load. The shearing f orce and the bending moment are related using the relationship below:
Where V is the shearing f orce and M is the bending moment.
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Equation 6
3.
Exp
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rim
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l App r '
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(
and
Pr c dur )
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Dial G auge 3.21 Apparatus
A
B
a
l
l
C
a
D
b
t E
RX
Knif e Edge Fixings
RY
W
W Figure 6: The S etup of Apparatus Conf iguration
The beam used f or the experiment was set up into a f our point loading f rame; Loads of weights W were hung at both sides of the beam. Th ere were reactions due to f ixings that were placed a distance of µa¶ f rom the Loads, such that they were equidistant f rom the loads; the f ixings were also equidistant f rom the dial gauge by a magnitude of µl¶ as shown in f igure 6. Theref ore the loading f rame was setup so that AB = CD, BE = EF and also the loads at A & D are both equal in magnitude. Measurements Taken AB =a =CD =0.167m; BE = l =EC = 0.125m, t (Thickness) = 0.0032m; b (Width) =0.025m 3.22 Procedure
Firstly, all the necessary dimensions of the solid beam were taken. This is highlighted in f igure 6 The Load W, on both ends of the beam was increased by 1Lb per reading, and the measurements were taken f or each of the columns of table 1.. Once the f irst f ive readings were taken, the Load W was now decreased by 1Lb. The dial gauge was used to measure the central def ection between b and c; a strain gauge bonded onto the solid beam f acilitated the measurement of the strain between b and c. Finally the Y oung¶s modulus was calculated using the gradient of the G raphs 1&2.
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3.3 Results and Analysis
Results 3.31 Table of Results The results obtained for the different quantities are shown in table 1. The quantities were measured in Lb and mm and were then converted to N and m using the conversions shown below: 1Lb = 4.448 N; Table 1: Empirical Results 1mm = 1 x 10 -3 m
Actual Recorded Results Strain
Results after Unit Conversions
No Of Readings
Weight (Lb)
Central Deflection X 10-2 mm
Weight (N)
1
1
46
100
4.448
2
2
96
200
8.896
3
3
142.5
295
4
4
190
5
5
6
Strain
Difference Columns
Central Deflection (m)
Difference in Weight (N)
Difference in Strain
0.0001
0.00046
0
0
0
0.0002
0.00096
4.448
0.1
0.0005
13.344
0.000295
0.001425
8.896
0.195
0.000965
390
17.792
0.00039
0.0019
13.344
0.29
0.00144
236
480
22.24
0.00048
0.00236
17.792
0.38
0.0019
4
191
390
17.792
0.00039
0.00191
13.344
0.29
0.00145
7
3
143
300
13.344
0.0003
0.00143
8.896
0.2
0.00097
8
2
97
200
8.896
0.0002
0.00097
4.448
0.1
0.00051
9
1
46
100
4.448
0.0001
0.00046
0
0
0
10-6 X
Difference in Deflection(m)
3.32 Graphs of Load against Strain and Central Deflection The graphs were plotted from the difference columns to eliminate initial condition errors. After the graph was plotted; the µline of best fit¶ and its algebraic function was obtained using the tools provided by Microsoft Excel. 20
Lo
18
d vs St in
16
14
Lo
0
y = 46548x - 0.134
12 d(N) 10 8
6 4
2 0
0
0.00005
0.0001
0.00015
St in0.0002 0
0.00025
1
Graph 1: Load vs Strain Page | 8
0.0003
0.0003 5
0.0004
The equation of the µline of best fit¶ for graph 1 was found to be y = 46548x - 0.1349
Graph 2: Load vs Central Deflection
Load
20
v Cent al Def lection
18
y = 9348.x - 0. 126
16 14
12 Load (
2
)10 8
6 4
2 0 0
0.0002
0.0004
0.0006
0.0008
C nt 3
4
0.001
0.0012
al Def lectio n
0.0014
0.0016
0.0018
0.002
(m), h
The equation of the µline of best fit¶ was found to be y = 9348.1x - 0.1266
Analysis
3.33 Calculation of Shearing Force and Bending Moment Moment A = Moment B = 0Nm [Free End] and the beam is in static equilibrium
Rx + Ry ± 2W =0 Ry = 2W ± Rx >>> Rx = 2W-Ry ««««««««««««««««««. Assuming the moment at A is equal to 0 and using 1:
Ry = W, Hence Rx = W can be deduced from 1.
A diagram of shear force against length of beam is shown figure 7.
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1
CD
W W
BC
Figure 7: Shearing Diagram AB
The bending moment between b& c doesn¶t change as V = 0N. 3.34 The Y oung¶s Modulus Calculation The Y oung¶s Modulus could not be directly calculated f rom the slopes of the graphs: the relationships seen in equations 2 and 3 had to be employed. From equation 3;
The bending moment between B and C (S ee section 3.33) is constant; theref ore M between B and C = Wa. y= t/2, f or a beam of unif or m cross section.
mation above and equation 2 the relation shown in equation was obtained in the manner Using the inf or below;
But
=G radient of graph 1
183 GPa
In order to verif y the above value, the Y oung¶s modulus was recalculated using the gradient of graph 2 in the f ollowing manner;
, where h is the central def lection
From equation 4,
Using equation 2&3 and substituting in f or
the equation below can be obtained:
179 GPa
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The values obtained f rom both the calculations maintained their accuracy. The theoretical result is 200GPa 3.35 U ncertainties in the measurement
When the diff erent measurements of the solid beam were taken, the measurements did not comply with the specif ication to meet the conditions described in section 3.33. AB & were CD not equal, their values diverged by a value of 4mm. B ut if this is the case the shearing and bending moments not be the same as shown in section 3.33. the graphs 1&2 the f ollowing were observed: Also when the algebraic equation of When Load = 0N, S train = -0.1349 & Central Def lection = -0.1266m These f actors were not theoretically possible f or a beam - in the elastic region ± and were considered to be experimental uncertainties. Th ese f actors will directly aff ect the gradient of the lines. 5
4.
Co nclusion
The laboratory session helped to f acilitate a thorough understanding of the elastic beam theory and the diff erence in the calculations of stress and strain when an axial f orce is applied and when a f orce per pendicular to the beam is applied. U navoidable mistakes, measuring uncertainties and small f aults in the apparatus may have contributed to the discrepancy in the calculated values; but they were not f ar f rom the theoretical value.
5. References 1. h ttp://www.efunda. com/designstandards/sensors/strain_gages/strain_gage_sensitivity.cf m, Date Accessed 13/2/2011 2. La boratory Boo klet, Gauge Se le ction Parameters. 2011 3. Bird, J. Ross, C., 2002, Me ch ani ca l Engineering Prin ciples, 1st Edition, Oxf ord nd 4. Hanna , J. Hi ller, M.J., App lied Me ch ani cs, 2 edition, 1967, London h 5. h tt //www.cir cuitstoday.com/strain -gauge, Accessed on 131/11 p:
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