Download Strain Gauges 1-Mechanical Engineering-Simulation Experiments and more Exercises Mechanics in PDF only on Docsity!
ABSTRACT
In this experiment we learned how to use strain gauges to measure different parameters such as Elastic Modulus, Poisson Ratio, Stress concentration factor (SCF) and Principal stresses
developed in materials due to loading. The results obtained are tabulated in tables.
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 and this will increase its electrical resistance end-to-end. Conversely, when a conductor is compressed such that it does not buckle, it will broaden and shorten and this will 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
THEORY PORTION
- 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 formulae which will be used throughout the experiment are written in respective section of experiment.
- 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 determine 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.
- 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 Wheatstone bridge while 03 bridge completion resistors to be internally provided by the indicator), half bridge (two gage acting as 02 arms of the Wheatstone bridge while 02 bridge completion resistors to be internally provided by the indicator; strain displaced by the indicator is equal to the algebraic difference of the strain sensed by the gages) and full bridge (04 gage acting as the arms of the Wheatstone bridge- the full bridge is however not a part of our experiment).
Following steps are followed to use strain indicator:
- After making proper connections of the wire the function switch is first placed at ZERO position for confirming instrument zero.
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 )
- Stress and Strain concentration:
Beam used in this part of experiment had hole in it to determine the stress concentration 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
- 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 “ Hooke’s Law”
- Constant Stress Beam
Load was applied to variable area or constant stress beam to produce deflection in beam. Strains values ε1, ε2, ε3, ε 4 were noted from indicator. Calculations were then performed to determine the values of ka, kb, and εa, εb. using following formulas
For the region A of the beam
ba(x) = kax εa = 0.5 (ε 1 + ε 2 ) x 1 = 220 mm, b 1 = 21.6 mm, ka = b 1 /x 1
For the region B of the beam
bb(x) = kbx εb = 0.5 (ε 3 + ε 4 ) x 3 = 117 mm, b 3 = 23.4 mm, kb = b 3 /x 3
OBSERVATIONS & CALCULATIONS
1) Determination of Young’s Modulus
L = 260 mm b = 25 mm d = 3.06 mm σ(L) = (6PL)/(bd^2 )
TABLE 01 – Stress & Strain Data
Obs. # P (gram force) P (N) Strain (μ) Stress ‘σ’ (MPa)
∆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.
Hence Poisson’s Ratio of the beam comes out to be 0..
3) Principal Stresses & Strains
L= 260 mm b = 25 mm d = 3.14 mm P = 500 gram force
TABLE 03 – Data for calculation of Principle Stresses & Strains
Load (gram force) 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 ) = 0.738 Ma
4) Stress & Strain Concentration
L 1 = 6.75 ”
L 2 = 9 ”
b 1 = 1 ” – 0.25 ” = 0.75 ” b 2 = 1 ”
TABLE 04 – Data for calculation of Stress & Strain Concentration
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 ) - 1.2C = -1.2x 10- A = ε 1 - 0.743 B - 0.552C = 1.73 x 10-
εmax = A + B + C = 2.33 x 10-
Incorrect Gage Factor x Observed Strain = Correct Gage Factor x Corrected strain CorrectedStrain = ε4c = 1.5x10- SCF = εmax /ε4c = 1.
5) Computation of the Applied Load through data of quarter & half bridges and Bending Stress calculations
b = 25 mm d = 6.3 mm Distance of center of gage 1 from loading point X 1 = 260 mm X 1 – X 2 = 76 mm X 2 – X 3 = 76 mm
TABLE 05
P ε 1 ε 2 ε 3 ε 1 – ε 2 (Half Bridge)
ε 2 – ε 3 (Half Bridge) 1 kg force 224 153 89 71 64
