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Abstract:
This experiment is performed to get the know how of uses of strain gauges in mechanics. Main purpose of experiment is to under stand how to use strain gauges to measure the material parameters such as Elastic Modulus, Poisson Ratio, Stress concentration factor(SCF), and Principal stresses developed in materials due to loading. The Results obtained in experiment are tabulated in Observations and Calculations.
Apparatus:
Starin gauges mounted beams, Strain indicator, Flexture, Connecting wires.
Introduction:
A strain gauge is a device used to measure the strain of an object. It was Invented by Edward E. Simmons and Arthur C. Ruge in 1938, the most common type of strain gauge consists of an insulating flexible backing which supports a metallic foil pattern. The gauge is attached to the object by a suitable adhesive, such as cyanoacrylate As the object is deformed, the foil is deformed, causing its electrical resistance to change. This resistance change, usually measured using a Wheatstone bridge, is related to the strain by the quantity known as the gauge factor.
A strain gauge takes advantage of the physical property of electrical conductance and its dependence on the conductor's geometry. When an electrical conductor is stretched within the limits of its elasticity such that it does not break or permanently deform, it will become narrower and longer, changes that increase its electrical resistance end-to-end. Conversely, when a conductor is compressed such that it does not buckle, it will broaden and shorten, changes that decrease its electrical resistance end-to-end. From the measured electrical resistance of the strain gauge, the amount of applied stress may be inferred
Procedure:
This Experiment mainly consists of 6 parts. These are listed below.
Theory Portion
Strain Gage Theory THE WHEATSTONE BRIDGE The strain gage indicator
Experimental Portion
Determination of Young’s Modulus (E) Determination of υ Poisson’s Ratio. Principal Strains and Stresses Stress and Strain Concentration Computation of the Applied Load through data of quarter & half bridges and Bending Stress calculations Constant Stress Beam
1.Strain Gauge Theory:
Strain gauge theory is similar to introduction of experiment in which basic working and purpose of strain gauge is discussed. Additionally some formulaes which will be used throughout the experiment are written in respective section of experiment.
2.Wheatstone Bridge:
Wheatstone bridge is a circuit which is used to determine the change in resistance of gauge when it is subjected to strain. Wheatstone bridge can be used to detemine both dynamic and static gauge readings.
The wheatstone bridge can be used in two ways:
As a direct-readout device, where the output voltage ΔE is measured and related to strain. As a null-balance system, where the output voltage ΔE is adjusted to a zero value by adjusting the resistive balance of the bridge.
3.Strain Gauge Indicator:
The strain gage indicator is a device which indicates the strain produced in form of digits. It provides diagrams for proper connections of the wires at the front panel of the indicator. These diagram are for quarter bridge (one gage acting as an arm of the
A graph between Axial and transverse strain was then plotted. Following equations were used to determine theaxial and transverse strains nad poissons ratio
∆εa = (1- o Kt) (∆ε’a - Kt ∆ε’t ) / (1 – K^2 t)
∆εt = (1- o Kt) (∆ε’t - Kt ∆ε’a ) / (1 – K^2 t)
= ∆εt / ∆ εa
3. Principal Strains and Stresses.
Rectangular Rossette was used to determine these parameters Rossette was set in such a way that centre lines of the gages of rectangular rosette of our experiment do not coincide with either of the actual principal directions. After mounting the rosette on beam, the known load was applied to measure three different strains those were εa, εb, εc. Calculations were then performed by puttin the values of strains in following formulas to determine Principal stresses.
` Major principal strain = ε 1 = 0.5 (εa + εc) + 0.5 *(εa – εc)^2 + (2εb – εa – εc)^2 ]0.
Minor principal strain = ε 2 = 0.5 (εa + εc) - 0.5 *(εa – εc)^2 + (2εb – εa – εc)^2 ]0.
tan 2θ = (2εb – εa – εc) / (εa – εc)
σ 1 = E (ε 1 + υε 2 ) / (1- υ^2 )
σ 2 = E (ε 2 + υε 1 ) / (1- υ^2 )
4. Stress and Strain concentration:
Beam used in this part of experiment had hole in it to determine the stress concentartion factor. Gauge was mounted on beam just at the end of hole, hence extrapolation technique was used to determine the SCF at center of hole. Same load was applied to introduced the deflection in beam. That deflection produced strains in beam which were indicated by strain indicator. Those strains were denoted as ε1, ε2, ε3, ε4.
Calculations were then performed to determine the principal stressesusing these formulas.
ε = A + B (R/x)^2 + C (R/x)^4
C = 5.86 (ε 1 – ε 2 ) – 5.44 (ε 2 – ε 3 )
B= 3.49 (ε 1 – ε 2 ) - 1.2 C
A = ε 1 - 0.743 B - 0.552 C
εmax = A + B + C
Stress concentration correction formula was also applied because incorrect gauge factor was used to determine ε4.
Incorrect Gage Factor x Observed Strain = Correct Gage Factor x Corrected strain Finally this formula was used to determine SCF
SCF = εmax /ε4c
5. Computation of the Applied Load through data of quarter &
half bridges and Bending Stress calculations
Known load was applied to introduce deflection in beam. Strains were indicated on strain indicator, and the difference between strains were confirmed using half as well as quarter bridge circuit. Calculations were then performed to determine applied load, ad bending stresses using following formulas.
P = [E bd^2 /6] [(ε 1 – ε 2 ) / ( X 1 – X 2 )]
P = [E bd^2 /6] [(ε 2 – ε 3 ) / ( X 2 – X 3 )]
σ 1 = 6PX 1 /bd^2 Bending Formula
σ 1 = Eε 1 Hook’s Law
Graph # 1: Stress vs Strain
Slope of graph is 71.
Hence young’s Modulus of material is = 72 Gpa.
Determination of υ Poisson’s Ratio.
L = 260 mm
b = 25 mm
d = 6.3 mm
o = 0.285 K = 0.
Table # 2.0: Poisson Ratio
P (gram force) ε’a (μ) ε’t (μ) 500 107 34 600 129 40 700 151 47 800 172 54 900 194 61 1000 215 67 ∆εa = (1- o Kt) (∆ε’a - Kt ∆ε’t ) / (1 – K^2 t) = 22 μ
∆εt = (1- o Kt) (∆ε’t - Kt ∆ε’a ) / (1 – K^2 t) = 7 μ
= ∆εt / ∆ εa = 0.
30
35
40
45
50
55
60
65
70
400 500 600 700 800 900 1000
Strain(μ)
stress(Mpa)
Principal Strains and Stresses
L= 260 mm, b = 25 mm, d = 3.14 mm, P= 500 gram force
Table # 3.
Load(gm f) εa(μ) Εb (μ) Εc (μ) 500 311 420 10 Major principal strain = ε 1 = 0.5 (εa + εc) + 0.5 *(εa – εc)^2 + (2εb – εa – εc)^2 ]0.5^ = 460 μ
Minor principal strain = ε 2 = 0.5 (εa + εc) - 0.5 *(εa – εc)^2 + (2εb – εa – εc)^2 ]0.5^ = -138 μ
tan 2θ = (2εb – εa – εc) / (εa – εc) => θ = 29.9o
σ 1 = E (ε 1 + υε 2 ) / (1- υ^2 ) = 40 Mpa
σ 2 = E (ε 2 + υε 1 ) / (1- υ^2 ) = 22.3 Mpa
Stress and Strain Concentration
L 1 = 6.75 inch, L 2 = 9inch, b 1 = 1- 0.25 = 0.75 inch , b 2 = 1 inch.
Table # 4.
Deflection(mm) 0. ε 1 (μ) 184 ε 2 (μ)^157 Ε 3 (μ)^149 Ε 4 (μ)^153 C = 5.86 (ε 1 – ε 2 ) – 5.44 (ε 2 – ε 3 ) = 1.8 x 10-
B= 3.49 (ε 1 – ε 2 ) - 1.2 C = -1.2x 10-
A = ε 1 - 0.743 B - 0.552 C = 1.73 x 10-
εmax = A+B+C = 2.33 x 10-
Incorrect Gage Factor x Observed Strain = Correct Gage Factor x Corrected strain
Correctred Strain = ε4c = 1.5x10-
SCF = εmax /ε4c = 1.
For the region B of the beam
εb = 0.5 (ε 3 + ε 4 ) = 67 μ
x 3 = 117 mm, b 3 = 23.4 mm, kb = b 3 /x 3 = 0.
εa/εb = kb/ka = 2
Discussion
