Strain Gauge Lab and Young’s Modulus Measurement

Engineering Mechanics 2: 16232

Jeswin Mathew

200901475Electrical and Mechanical Engineering

ContentsIntroduction………………………………………………………………………………………….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

Page | 1

1. Introduction 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 ‘cleansed spot’ (The reasons for this will be discussed later) using a permanent adhesive. The second session entailed the measurement of the Young’s Modulus of a solid beam by applying a load on both ends of the bar and measuring the increments in strain (ε) and the central deflection (δ) 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. The Strain Gauge Experiment

2.1 Theoretical Background

2.11 Introduction

The primary component using in the manufacturing of a strain gauge 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, causing its 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]:

ε=dRROx GaugeFactor , [1] Equation 1

Where dR is the change in resistance due to deformation,

Ro is the resistance when unstressed.

Page | 2

Foil Grid

2.12 Types of strain sensitive alloys

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.

Constantan Alloy: I. This alloy possesses a very high strain sensitivity (Gauge Factor)

II. It is characterised by a good fatigue life and can handle relatively high elongations.III. It is suitable for the measurement of very large strains (5%).IV. 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 preparation 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]:

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.

Page | 3

Decrease Solder Tabs

Backing Material or encapsulation

Figure 1: Bonded Strain Gauge

Surface Abrading and marking out the layout lines: The surface abrading procedure was performed using a silicon carbide paper and metal conditioner solution, and was done to remove 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 the metal conditioner and was cleansed by rubbing the paper against the surface. . After drying, the layout where the strain gauge was to be bonded was marked using a ball point pen with a pair of crossed, perpendicular reference lines.

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

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.

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.

Page | 4

Figure 2: Surface Abrading

Figure 3: Strain Gauge removed from glass slab.

Figure 4: Application of Adhesive

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 ResultsThe 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 121Ω which can be taken to be RO . It was also observed that there were only obscure differences in the electrical resistances when the beam was stressed.

3. The Measurement of the Young’s Modulus

3.1 Theoretical Background

3.11 The Elastic Beam Theory

If a beam, portraying elastic properties, of symmetrical cross-section is subject to a bending moment M, then stresses will occur along the surface and also the beam will bend into a simple arc as shown in figure 5. It can be noted from the figure that the upper fibres of the beam are in tension due to the increase in their length and the bottom fibres will be in compression due to the converse occurring. For a beam subject to bending moments as seen in the figure, the stress and the Young’s modulus can be related using equation [4] 2:

E=σε

Where, E is the Young’s modulus of the material in N/m2,

σ is the stress and is given by the equation Force/ Area, its units are N/m2,

Page | 5

Radius of Curvature of the Neutral Axis, ‘R’

The Distance ‘y’ from Neutral Axis; +y for above the axis and –y for below

Equation 2

Figure 5: Bending of Beams [3]

MM

ε is the strain and is given by Elongation/Original Length.

Another relation that encapsulates, the Young’s modulus, stress and strain is shown in equation 3:

σy=MI

= ER

Where M is the moment in Nm,

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 from 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 σ andε experienced by the beam – caused by bending moments as shown in figure 5 - increases linearly as y increases i.e. the maximum strain is observed at the top surface of the beam which has the greatest distance from the neutral axis. The strain increases linearly and is related to y by the radius of curvature (R) for the neutral layer using the equation below [4]:

ε=± yR

Also the stress at distance y from the neutral axis can also be calculated from the above formula by using the relationship portrayed in equation 2:

σ=±E yR

The ± notation beside y indicates that y is positive for distances above the neutral axis and negative for below the neutral axis as shown in figure 5. The neutral axis for a beam of uniform cross-section passes right through the centroid of its cross-section. The radius of curvature is a quantity that is practically impossible to obtain an absolute value and therefore can only be approximated; when the beam is unstressed, the radius of curvature is undefined.

3.13 Shearing Force and Bending Moment

The shearing force in a beam at any section is the force transverse tending to cause it to shear across the section; the shear force will remain uniform on an unloaded part and will change abruptly at a concentrate load. The shearing force and the bending moment are related using the relationship below:

dMdx

=V

M=∫V dxWhere V is the shearing force and M is the bending moment.

Page | 6

Equation 3

Equation 4

Equation 5

Equation 6

E

3.2 Experimental Apparatus and Procedure

3.21 Apparatus

Configuration

The beam used for the experiment was set up into a four point loading frame; Loads of weights W were hung at both sides of the beam. There were reactions due to fixings that were placed a distance of ‘a’ from the Loads, such that they were equidistant from the loads; the fixings were also equidistant from the dial gauge by a magnitude of ‘l’ as shown in figure 6. Therefore the loading frame 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 figure 6 The Load W, on both ends of the beam was increased by 1Lb per reading, and the measurements were

taken for each of the columns of table 1.. Once the first five readings were taken, the Load W was now decreased by 1Lb. The dial gauge was used to measure the central defection between b and c; a strain gauge bonded onto

the solid beam facilitated the measurement of the strain between b and c. Finally the Young’s modulus was calculated using the gradient of the Graphs 1&2.

Page | 7

l ala

Figure 6: The Setup of Apparatus

A B C D

WW

RX RY

tb

Dial Gauge

Knife Edge Fixings

3.3 Results and Analysis

Results3.31 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;1mm = 1 x 10-3 m

Actual Recorded Results Results after Unit Conversions Difference Columns

No Of Reading

s

Weight (Lb)

Central Deflection X 10-2 mm

StrainX 10-6

Weight (N) Strain Central Deflection (m)

Difference in Weight

(N)

Difference in Strain

Difference in Deflection(m)

1 1 46 100 4.448 0.0001 0.00046 0 0 02 2 96 200 8.896 0.0002 0.00096 4.448 0.1 0.00053 3 142.5 295 13.344 0.000295 0.001425 8.896 0.195 0.0009654 4 190 390 17.792 0.00039 0.0019 13.344 0.29 0.001445 5 236 480 22.24 0.00048 0.00236 17.792 0.38 0.00196 4 191 390 17.792 0.00039 0.00191 13.344 0.29 0.001457 3 143 300 13.344 0.0003 0.00143 8.896 0.2 0.000978 2 97 200 8.896 0.0002 0.00097 4.448 0.1 0.000519 1 46 100 4.448 0.0001 0.00046 0 0 0

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 ‘line of best fit’ and its algebraic function was obtained using the tools provided by Microsoft Excel.

0 0.00005 0.0001 0.00015 0.0002 0.00025 0.0003 0.00035 0.00040

2

4

6

8

10

12

14

16

18

20

f(x) = 46548.0275516594 x − 0.134909204758922

Load vs Strain

Strain

Load (N)

Page | 8

Table 1: Empirical Results

Graph 1: Load vs Strain

The equation of the ‘line of best fit’ for graph 1 was found to be y = 46548x - 0.1349

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.0020

2

4

6

8

10

12

14

16

18

20

f(x) = 9348.11817691188 x − 0.126632677601491

Load vs Central Deflection

Central Deflection (m), h

Load (N)

The equation of the ‘line 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

∑ Fy=0Rx + Ry – 2W =0

Ry = 2W – Rx >>> Rx = 2W-Ry ……………………………………………….

Assuming the moment at A is equal to 0 and using 1:−W (2a+2 l )+R y (2 l+a )+R x (a )=0 ;

−2wa+2wl+2R yl+R y a+(2W−R y )a=0Ry = W,

Hence Rx = W can be deduced from 1.

A diagram of shear force against length of beam is shown figure 7.

Page | 9

CD

1

W

Graph 2: Load vs Central Deflection

The bending moment between b& c doesn’t change as V = 0N.

3.34 The Young’s Modulus Calculation

The Young’s Modulus could not be directly calculated from the slopes of the graphs: the relationships seen in equations 2 and 3 had to be employed.

From equation 3;

σ=MyI

The bending moment between B and C (See section 3.33) is constant; therefore M between B and C = Wa.y= t/2, for a beam of uniform cross section.

I= 112b t 3

Using the information above and equation 2 the relation shown in equation was obtained in the manner below;

E=MI ε

= 6ab t2 (

Wε )

But (Wε ) =Gradient of graph 1

E=46548 X ( 6 X 0.167

0.025 X0.00322 )E≅ 183 GPa

In order to verify the above value, the Young’s modulus was recalculated using the gradient of graph 2 in the following manner;

R≅ l2

2h , where h is the central deflection

From equation 4, ε=±( t2 )( 2hl2 )≡( tl2 )hhhhh Using equation 2&3 and substituting in for ε∧σ the equation below can be obtained:

E=6 al2

b t 3x (Wh

)

E=9348.1 X ( 6 X 0.167 x 0.12520.025 X 0.00322 )Page | 10

BC

AB

WFigure 7: Shearing Diagram

E≅ 179 GPaThe values obtained from both the calculations maintained their accuracy. The theoretical result is 200GPa

3.35 Uncertainties in the measurement

When the different measurements of the solid beam were taken, the measurements did not comply with the specification to meet the conditions described in section 3.33. AB & CD were not equal, their values diverged by a value of ± 4mm. But if this is the case the shearing and bending moments not be the same as shown in section 3.33.

Also when the algebraic equation of the graphs 1&2 the following were observed:When Load = 0N, Strain = -0.1349 & Central Deflection = -0.1266m

These factors were not theoretically possible for a beam - in the elastic region – and were considered to be experimental uncertainties. These factors will directly affect the gradient of the lines.

4. Conclusion

The laboratory session helped to facilitate a thorough understanding of the elastic beam theory and the difference in the calculations of stress and strain when an axial force is applied and when a force perpendicular to the beam is applied. Unavoidable mistakes, measuring uncertainties and small faults in the apparatus may have contributed to the discrepancy in the calculated values; but they were not far from the theoretical value.

5. References1. http://www.efunda.com/designstandards/sensors/strain_gages/strain_gage_sensitivity.cfm, Date

Accessed 13/2/20112. Laboratory Booklet, Gauge Selection Parameters. 20113. Bird, J. Ross, C., 2002, Mechanical Engineering Principles, 1st Edition, Oxford4. Hannah, J. Hiller, M.J., Applied Mechanics, 2nd edition, 1967, London5. http://www.circuitstoday.com/strain-gauge, Accessed on 131/11

Page | 11