Lab 5 Combined Loading

Circular Beam Under Combined Loading Nick Leach Group 3a MAE 244, Sec. 2, Dr. Feng November 9, 2005

Table of Contents: Introduction.......................................................................................................3-4 Schematics.........................................................................................................5-6 Analysis of Results............................................................................................7 Discussion..........................................................................................................8-11 References..........................................................................................................12 Appendix............................................................................................................13-17

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Introduction: The combined loading experiment is pertinent for engineering applications in strain gages. Strain gages are used in a variety of industrial applications for experimental stress and failure analysis and diagnosis on machinery such as impact, dental and medical sensors, web tension and tension sensors, force measurements in machine tools such as hydraulic and pneumatic presses, aerospace, automotive, and biometrics. Experiments are routinely performed in the laboratories in order to determine factors like residual stress, proof testing, and also measurements of vibration, torque, bending, deflection, compression and tension, and strain. In many engineering situations, components frequently have to withstand more than one type of load. Shafts often have to withstand torque and bending moments. To solve this type of design problem the components are assumed to behave in a linear manner and superposition is used (Yielding cannot occur). The stresses due to each type of loading are determined in turn and then combined using appropriate equations or Mohr's circle. Mohr’s circle is used in this lab to display the graphical approach to the solutions of the stress and strain values and also the angle. Introduced by Otto Mohr in 1882, Mohr's Circle illustrates principal stresses and stress transformations via a graphical format. The normal stesses equal the principal stresses when the stress element is aligned with the principal directions, and the shear stress equals the maximum shear stress when the stress element is rotated 45° away from the principal directions. The 2D plane stress diagram (in Figure 2) is used to draw Mohr's Circle.

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Under combined loading for a circular beam, each applied load is analyzed based on measured strains and principal directions, stresses and strains are determined at the location of the rosette using calculations and Mohr's circle. To obtain the calculations, the correct procedure should be followed and ensured. The location of each rosette and the directions of all the individual gages should be recorded. The gages should be connected to separate channels of the switch and balance unit (beside the strain indicator in the schematic), which in turn should be connected to the strain indicator. Then calibration is obtained when each gage circuit is balanced with no load. After this preliminary setup, the loads are applied and the strain readouts are recorded from the strain indicator. Experimental results are then calculated and comparisons are made with analytical values for error analysis.

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Schematic:

Figure 1: Circular Shaft with Fixed Support

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Figure 2: 2D Plane Stress Diagram (Ref. 4)

Figure 3: Mohr's Circle Diagram (Ref. 4)

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Analysis of Results:

εp =

θ=

(

)

ε xy 1 tan −1 2 εx −εy θ = principal strain direction

σp =

R=

εx + εy 1 (ε x + ε y ) 2 + ε xy 2 ± 2 4 εp = principal strain εx = x Cartesian strain component εy = y Cartesian strain component εxy = ½ Cartesian coordinate of engineering shear strain

σx +σy 1 (σ x + σ y ) 2 + σ xy 2 ± 2 4 σp = principal stress σx = x Cartesian stress component σy = y Cartesian stress component σxy = Cartesian shear stress component 1 (σ x + σ y ) 2 + σ xy 2 4 R = radius of circle

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•

Comparisons:

LOAD εX εY εZ γXY εXY εmax εmin θP 400 -67 227 246 332 166 301.73182 -141.73182 -24.236878 800 -134 452 492 666 333 602.5515754 -284.5515754 -24.32805086 Table 1: Calculated Experimental Results for Strain LOAD 400

σX -37.04552899

σY 6572.256797

σXY 3718.4

800

-92.45550824

13081.1879

7459.2

σmax 8252.2744 6 16465.680 7

σmin -1717.06319 -3476.94829

Table 2: Calculated Experimental Results for Stress LOAD 0 400 800

E 29000000 29000000 29000000

G 11200000 11200000 11200000

υ 0.29 0.29 0.29

M

T

0 2600 5200

0 2200 4400

Table 3: Calculated and Given Values for Data Analysis LOAD σX σY σXY 400 0 7601.095981 3215.8483 800

0

15202.19196 6431.6966

σmax 8779.0864 6 17558.172 9

σmin -1177.99

θP -20.11817

εmax 314.5070242

-2355.98

-20.11817

629.0140484 -256.8224

Table 4: Calculated Analytical Results for Stress and Principal Strains LOAD 400

σmax σmin 6.000761 45.76205964

θP 20.472523

800

6.222129 47.57964274

20.925709 4

εmax 4.0619773 9 4.2069764 8

εmin 10.3733874 10.79700121

Table 5: Percent Difference for Experimental/Theoretical Values 8

εmin -128.4112

Discussion: •

Conclusions: There are several ways to determine the principal stresses/strains, principal

direction, and maximum shear stress at locations of a structure or part under combined loadings using the three-element rosette strain gage. For the graphical approach, Mohr’s circle is an obvious choice of method. First you calculate the stresses and tau, then you can measure the angle and maximum and minimum values from the illustration. You don’t need to calculate these beforehand. Other than Mohr’s circle, using the calculations provided in the analysis of results can give direct values for the maximum and minimum stresses/strains and the angle too.

Figure 1: Mohr’s Circle for 400 lbs

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Figure 2: Mohr’s Circle for 800 lbs

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•

Limitations and Experimental Error:

Limitations to the experiment are mainly in the calculations. The moment equations used are a little off and effect the sigma values directly, thus increasing percent difference as a result of inaccurate analytical calculations. Since the strain measurements were in microstrain, then the error from the strain indicator is minimal due to the fact that the values are so high. Obviously, the percent difference table reflects some major problems in the comparisons and those problems mainly originate in the value conversions of units. As you can see, the applet values of theta are about .025% deviating from the values calculated by the data on Excel. As for the analytical calculations of the minimum principal stress, the error is completely in the calculations using the torque and moment. I believe it was an error on behalf of the moment of inertia (I) and the polar moment of inertia (J). Other than these errors, the experiment was smooth and I think it can be duplicated with similar experimental values.

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References: 1. Handout material supplied by instructor 2. www.engin.umich.edu/students/ support/mepo/ELRC/me211/mohr.html 3. Mechanics of Materials 6th Edition, R.C. Hibbler 4. www.efunda.com/formulae/solid_mechanics/mat_mechanics/mohr_circle.cfm 5. www.aoe.vt.edu/~jing/MohrCircle.html 6. www.blh.de/products/straingage.htm

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Appendix: Circular Shaft Dimensions Distance from Arm to Strain Gage: Y=6.5” Moment Arm Length: L=5.5” Shaft Diameter: D=1.516” Shaft Radius: R=0.758” Strain Gage R=350±0.2% Ω Sg=2.045±0.5% Applied Load (lbs) 0 400.0 800.0

Strain Gage at A 1 (0) -66 (-67) -133 (-134)

Strain Gage at B 0 227 452

Strain Gage at C 1 (0) 247 (246) 493 (492)

Appendix Table 1: Original Recorded Data from Experiment

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