Lab 4 - Stress Analysis of Beams Using Strain Gauges PDF

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Lab 4: Stress Analysis of Beams Using Strain Gauges Section ML-X March 10, 2017 2 Objective: The purpose of this experiment is to determine the distributions of stress and strain produced from pure bending in a beam using strain gauges. The shear modulus, Poisson’s ratio and the elastic modulus are ...

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Lab 4: Stress Analysis of Beams Using Strain Gauges Section ML-X

March 10, 2017

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Objective: The purpose of this experiment is to determine the distributions of stress and strain produced from pure bending in a beam using strain gauges. The shear modulus, Poisson’s ratio and the elastic modulus are established through these strain measurements. Introduction: A material is deemed to be in bending if it is subjected to compressive stresses on one portion of a cross section and tensile stresses on the other portion. Pure bending is achieved under particular conditions where there is usually transverse shear present. Applying a moment that is equal and opposite in direction on each side of the beam will cause bending. The bending in a beam is commonly known as “flexure”. In plane bending, where plane portions also remain plane after bending, strain is proportional to the distance from the neutral axis. Based on whether a force is tensile or compressive, a solid subjected to uniaxial force (will deform in the direction of that force), will also expand or contract laterally. The lateral and axial strains will maintain a constant relationship if the material stays elastic under the applied force, and if the material is homogeneous and isotropic. Poisson’s ratio can be defined as this constant relationship. When a force is applied to a certain material, this constant can be used to predict the change in length of that material. The right material can then be established for practical implications of a structure that will be subjected to any type of deformation.

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Procedure: 1. Measure the cross section of the steel bar and record the position of each strain gauge. 2. Ensure that the beam is symmetric with respect to the centerline of the strain gauges and do not apply a load. 3. Press the power button on the strain indicator to turn it on and switch the gauge selector to channel 1. 4. Ensure that the gauge strain indicator reads zero or close to zero when there is no load applied by adjusting the potentiometer. Do the same for channels 2 to 6. 5. Apply a load of 1000N every 1000N until 5000N is reached, recording the strain for each channel (6) at each increase. 6. Unload the beam and record the strain at zero load to verify (should be approximately zero). Results: 1. Moment of inertia of beam: I [mm4] = bh3/12

b = 0.01904 m 3

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4

I = (0.01904m)*(0.03192m) / 12 = 5.1602928x10 m 2. Free body diagram:

h = 0.03192 m

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Shear and bending moment diagrams for an arbitrary load:

3. Bending stresses at strain gauge points: Ex for channel #1, P = 1000N: M = (P*d/2) = (1000N/2)*0.1925m = 96.25 Nm σ = My/I Where M = moment, y = distance from neutral axis, I = moment of inertia = (96.25 Nm*0.01596m)/(5.1602928x10-8 m4) = 2.977x107 Pa

Bending stress for each strain gauge in Pa:

The negative values indicate compression and the positive values indicate tension.

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4. Stress-strain curves for gauge points 1 – 5 and elastic modulus (E):

Bending Stress (Pa

Stress-Strain Curves 160000000 140000000 120000000 100000000 80000000 60000000 40000000 20000000 0 0

2

4

6

8

10

Strain

E = σ/ = 189.653 GPa 5. Longitudinal strain, transverse strain:

Load vs. Strain for Gauge 5 & 6 0.01 Strain (microstrain)

0.01 0.01 0 0 0 0

0

2

4

6

8

0 Load (N)

Poisson’s ratio: Slope for transverse strain is: �T/�P = 4.325x10-7 Slope for axial strain is: �L/�P = 1.5575x10-6

10

12

12

6 Poisson’s ratio: v = (�T/�P)/(�L/�P) = 0.278 6. Stress at each gauge point: σ = E where E=200x109 Pa Stress (Pa): Load (N) 1000 2000 3000 4000 5000

1 -35000000 -62600000 -93600000 -125400000 -158200000

2 -20400000 -37200000 -56400000 -76200000 -95800000

3 0 600000 1200000 1800000 2400000

4 22000000 39800000 59800000 69800000 100000000

5 34400000 62600000 94400000 126600000 159000000

7. Shear modulus: G = E/[2(1 + v)] = (189.653 GPa)/[2*(1+0.278)] = 74.20 GPa Compared to the calculated shear modulus of steel of 77 GPa, 74.20 GPa is under the published value but is also close, and gives a percent error of only 3.64%. 8. Strain vs. distance:

Strain (microstrain)

Strain vs Distance for 5000N 0.01

-20

-15 0

0.01 0.01 0.01 0.01 0 0 0 0 0 0 -10 -5 0 -0.01 -0.01 -0.01

5

10

15

20

0 -0.01

Distance from horiztonal mid-plane of beam

The neutral axis found from the graph lies at approximately 0.1mm above the theoretical neutral axis, 0. This is most likely due to the direction of the applied force on the beam.

6 -9200000 -17000000 -25600000 -34800000 -43800000

7 Report: 1. Loa

Theoretical and Experimental Stress (MPa)

d

1

2

3

4

5

(N) Theory

Exp

Theory

Exp

Theory

Exp

Theory

Exp

1000 2000 3000 4000

-35.00 -62.60 -93.60 -125.40

-29.77 -59.54 -89.31 -119.07

-20.40 -37.20 -56.40 -76.20

-18.84 -37.68 -56.52 -75.35

0 0.60 1.20 1.80

0 0 0 0

22.00 39.80 59.80 69.80 100.0

18.84 37.68 56.52 75.35

5000

-158.20

-148.84

-95.80

0

0

Compression

-94.19 Compression

2.40 Tension

Theory

Exp

34.40 29.77 62.60 59.54 94.40 89.31 126.60 119.07 159.00

94.19 Tension

148.84 Tension

The values indicate that the top surface of the beam where gauge 1 is located is under a compressive stress laterally, as well as the 2nd gauge but with a compressive stress of less intensity. The stress at the neutral axis is zero. Gauge 5, placed under the beam, is under tension from the force.

Discussion: 1. The experimental value of the neutral axis from the plotted graph of stress vs. strain is very close to 0. Even though we did not take into account the beam’s strain values after unloading, it is said that in plane bending, where the plane

8 sections also remain plane after bending, strain is proportional to the distance from the neutral axis, which is the case for the 5000N load plotted graph. 2. The discrepancies between the theoretical and experimental stress values are due to the use of different formulas. The experimental values were calculated using the moment, the distance from neutral axis and the moment of inertia. The theoretical values of stress were calculated using the published value of the elastic modulus of 200x109 Pa multiplied by the experimental values of strain. Although these formulas use different methods of calculation, they come out to roughly close values. The main difference between the two stress values is the elastic modulus used (theoretical or experimental). The possible sources of error could be due to round off errors or mistake in or round off in measurements. 3.

Conclusion: To conclude, it is found that the beam felt tensional stress from the applied load on its bottom surface, and it felt compressive stress on its top surface from the load. From calculating and obtaining values for the moment of inertia of the beam, the moment and

9 the distance from the neutral axis, the bending stress could be calculated. Further, the elastic modulus, E, was obtained, from the stresses and strain values, at a value of 189.65 GPa, which is indeed close to the published value of 200 GPa. The Poisson’s ratio was also determined to be 0.278, which is very close to the theoretical value for steel of approximately 0.27 to 0.30. The shear modulus, G, was also calculated to be 74.20 GPa which compares with the published value of 77 GPa for steel. The data tables and graphs facilitated the visualization of the data and established the relationships between stress and strain, load and strain, and load and distance. They also show that the beam’s plane sections did remain plane after bending. Unused experimental data is the unloading strain values of each gauge. These values were to ensure that the strain went back to zero after loading. This test is crucial in establishing the most favorable materials to use for structures that will be subjected to any type of deformation....