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I. OBJECTIVES
- To compare the experimentally determined stresses in a T-beam with those predicted from the beam-bending stress theory.
- To compare the location of the neutral axis determined by: a. Hand calculation using the centroid equation b. The location determined from balancing a unit length of the cross section. c. Experimentally determined from the strain gage data.
- To use the data acquisition program to obtain measurements from electrical strain gages and to investigate the distribution of stress in the T-Beam.
- To familiarize the students with the LABTECH data acquisition program.
II. INTRODUCTION AND BACKGROUND
Electrical resistance strain gages are the most frequently used devices in stress-strain work throughout the world today. The electrical strain gage operates on the direct relationship between the change in electrical resistance of a wire as it is stretched and the strain εεεε developed within the material. The ability to precisely measure the change in electrical resistance gives a direct, precise measure of the strain.
As a wire is stretched, its length increases and its cross sectional area decreases, which increases the resistance of the wire. By bonding the strain gage to a structural member and measuring the change in resistance as the load is applied, the corresponding strain can be measured. The experimental value of stress σσσσ may be determined from the measured strain εεεε by using Hook’s Law for uniaxial stress, σσσσ = Eεεεε. E is the modulus of elasticity of the beam material. The beam used in this experiment is made of 7075 aluminum which has a modulus, E = 10x10 6 psi.
The theoretical maximum stress developed within a beam is calculated using the flexure equation over the linear region of stress and strain.
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Bending stress equation:
Mc
σ max = 2.
I
where:
M = the maximum moment
I = the moment of inertia about the centroidal axis
c = the maximum distance from the centroidal axis
The moment of inertia of a non-centroidal cross-sectional area about the centroidal axis is obtained by use of the parallel axis theorem.
I = I 0 + Ad 2 2.
where:
I (^) o = the moment of inertia about the centrodial axis
A = corresponding cross-sectional area
d = distance between centroid and any parallel axis
The location of the centroid is determined from the following equation:
n
∑ A i yi
_ i= y = 2. n
∑ A i
i=
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Once the voltage ratio V (^) r is found, the following equation is used to calculate the strain:
4Vr
GF * (1 + 2Vr)
where:
εεεε = strain
GF = constant gage factor of 2.
III. EQUIPMENT LIST
- 10,000 lb. compression machine
- Bridge Completion Unit
- 30 inch long, 7075 T6 aluminum T-beam with eight mounted strain gages
- Small length cut from T-beam (use for cross section measurements)
- Scale and micrometer
- LABTECH Program: Data acquisition instruction
- One 3 1/2 inch floppy disk
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IV. PROCEDURE
4.1. Measure and record all necessary dimensions of the T-Beam, load and support points. The location of all the strain gages in relation to the base of the beam are stored in a file called gageloc.xls in your instructor’s directory. Do not remove the beam from the test fixture as the strain gages are easily damaged.
4.2. Calculate the location of the centroidal axis and the moment of inertia. Check the result with your instructor. A small length of the T beam has been cut off and is provided so that the centroid can be checked by finding the balancing point of the cross section.
4.3. Calculate the maximum load P that can be applied to the beam given an allowable stress bending stress of 10,000 psi. Check your computation with the lab instructor before starting the experiment.
4.4. launch LABTECH by double clicking on the “LABTECH pro” icon. Double click on the “Build -Time” icon. Open the file (substitute your instructor’s name for “INSTRUCTOR”) “C:\NBP_WIN\INSTRUCTOR\NEWTB.LTC”.
4.5 With the beam unloaded run the LABTECH program to acquire the strain readings present in the unloaded beam. Click the LABTECH “Run” icon located on the left-hand side of the screen. Click “OK” to start the acquisition. Data will be saved in your instructor’s directory under the file name STRAIN&.XLS. The & character will be replaced with a number. Each time you acquire data this sequence number will increase by one (STRAIN1.XLS, STRAIN2.XLS etc.).
BUILD-TIME ICON IN PROGRAM MANAGER
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V. REPORT
5.1. Beam Properties
For the T-beam, include the tabulated beam information recorded in Table II along with a table of the position of the eight strain gages with reference to the bottom of the beam (Table III) in the Results section. Calculations for the moment of inertia, the location of the centroidal axis and the maximum allowable load, should be put in the Appendix.
5.2. Graphs and Calculations
Results should include a graph of the stress, calculated from σσσσ = E εεεε, versus distance from the bottom of the beam (tensile stress is positive and compressive stress is negative). From this graph, determine the location of the neutral axis. Plot on this same graph the corresponding theoretical stress curve (using equation 2.1). This graph, will thus show both the experimental and theoretical stress distribution, from the bottom of the beam to the top. Plot the same experimental and theoretical curves for the other two load cases (three graphs in total). If a strain gage in not functioning, omit its data point from the graph. Make sure you use an X-Y plot since the distance between the gages varies.
Figure 2 Example graph of Theoretical and Experimental Stress Results
STRESS
(PSI)
DISTANCE FROM BOTTOM OF BEAM (IN)
0
LOCATION OF NEUTRAL AXIS, σσσσ = 0
NEGATIVE STRESS, TOP OF BEAM
SHOW DATA POINTS
EXPERIMENTAL RESULT
THEORETICAL RESULT
POSITIVE STRESS, BOTTOM OF BEAM
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5.3. Comparison of Results
Compare the theoretical and experimental stress graphs for each load case. These three graphs should be presented in the Results section with titles and axis labels. Excel can be used to obtain a best linear curve fit of the data points. The equation of this linear curve fit has the form Y = mx +b which can be displayed on the graph (right click on the data points in the graph and use insert trendline and under options, select display equation on chart). For each graph compare the slope m and the neutral axis location x (x =- b/m when Y = 0). Make percent comparisons of these two factors for each load level. Put these results in a table and include it in the Results section. Raw data (for example, the spread sheet strain data) go in the Appendix. The Result section is the most important part of the report so take time to describe each table and graph that you are presenting (use Figure and Table numbers). Also include the value of the centroid found from balancing the cross section.
VI. SELECTED REFERENCES
6.1 Introduction to Mechanics of Solids, Popov, pp. 177-188.
6.2. Engineering Mechanics of Deformable Bodies, pp. 205-273.
6.3. Strain Gauge Techniques, Murray and Stein
6.4. Statics and Strength of Materials, Stevens, Ch. 8 and 11
6.5 Experimental Stress Analysis, Daily, James.
6.6 LABTECH Data Acquisition and Process Control Software, Windows Users Guide
Note: Major contributions to this experiment were made by:
HOOMAN NASTARIN, MASARU KENT KAWAI, JONGLIM KIM, CATHERINE
MICHAUD, LINDA MILES, GHASSAN SAKAKINE, MARIA SOTO-CARDENAS,
CALVIN WOO
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MECHANICS LABORATORY
AM 317
EXPERIMENT #3. STRESS MEASUREMENT IN A T-BEAM
STUDENT’S NAME: __________________________GROUP NO.: ________
INSTRUCTOR’S NAME: _______________________DATE OF EXP.: ______
n
∑ A i yi
i= y = n
∑ A i
i=
Centroid Calculation
Figure 1. Cross-Section.
SECTION bi hi Ai = bi x hi Yi Ai x Yi i = 1 BOTTOM i = 2 TOP ΣΑΣΑΣΑΣΑ i ==== ΣΑΣΑΣΑΣΑ iYi ====
Table I Calculation of the centroid of the cross-section.
Beam Specifications Data
Position of Neutral Axis, y (in.) Moment of Inertia, I (in.4) Length of the Beam, L (in.) Modulus of Elasticity, E (psi) Maximum Allowable Stress, σσσσ (psi)
Table II Beam Data
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Figure 2. Gage Locations
Gage No.
Position of Gage From Bottom of Beam
Table III Gage Position (Do not remove beam from support, get the gage locations from the excel file gageloc.xls)
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Appendix A Calibration of the Strain Gages Optional Method
Temperature changes and other effects can cause the strain gages to go out of calibration. This goal of calibration is to have the strain gages read zero strain when the beam is unloaded. Make sure that no load is present on the beam when calibrating the gages.
A.1 Launch LABTECH by double clicking on the “LABTECH pro” icon. Double click on the “Build time” icon. Open the file (substitute your instructor’s name for “INSTRUCTOR”) :
“C:\NBP_WIN\INSTRUCTOR\CALIB.LTC”.
A.2 Make sure the beam is unloaded and click the LABTECH “Run” icon located on the left-hand side of the screen. Click “OK” to start the acquisition. Data will be saved in your instructor’s directory under the file name CALIB&.XLS. The & character will be replaced with a sequence number. Each time you acquire data the number will increase by one (CALIB1.XLS, CALIB2.XLS etc.). The instructor should reset the counter back to 1 at the end of each class and delete old files). To reset the counter, double click on the file icon at the center.
A.3 Use Excel to open the CALIB&.XLS file you created. The LABTECH program has stored 60 measured voltage values (later converted to strain) for each of the eight strain gage as shown in Table 1 below (only two of the 60 voltage values are shown for each gage):
Table 1 Excel file containing calibration data
A.4. The first few and last few strain values may be inaccurate due to transients and will not be used to calculate the average voltage values. To calculate the average strain, go to cell A65 and enter the following equation: =AVERAGE(A5:A55)/3300. The 3300 value comes from the amplification of the strain gage amplifier.
GAGE 1 GAGE 2 GAGE 3 GAGE 4 GAGE 5 GAGE 6 GAGE 7 GAGE 8 1.262936 2.2299865 0.967050 3.492923 1.1149932 2.229986 3.344980 1. 1.514322 2.1740598 0.659738 3.688382 1.0870299 2.174060 3.261090 1.
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Copy the equation in cell A65 to cells B65 to H65 such that you have an average value for each column of data. Note these calibration values in Table 2:
Table 2 Record the calculated calibration values in this table
A.5. Return to the LABTECH program. Hold down the alt key and press the tab key until the LABTECH icon appears. Open the file: “C:\NBP_WIN\INSTRUCTOR\NEWTB.LTC”
A.6. Click ZOOM (left side of screen) to enlarge the flow diagram at the top center of the screen. Double-Click on the AI (analog input) icon labeled for strain gage 1. Make sure that Strain 1 is displayed in the title bar, if not, click cancel and try again until you find the analog input for gage 1.
GAGE 1 GAGE 2 GAGE 3 GAGE 4 GAGE 5 GAGE 6 GAGE 7 GAGE 8
ZOOM ON STRAIN 1
ICONS
ANALOG INPUT ICON
FOR GAGE 1
ZOOM ICON
