Shear and Moment DistributionInCantilever Beams

Team Emily DiMartiniDan QuirogaMonica SharobeamBrandon SimonSubmitted: 04/02/2015

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This (Evaluation) sheet. (Pages 2-3 of this evaluation completed/attached by the team as a self-assessment/ checklist) Rating and Assessment Sheet. Abstract.....Numbering of Pages.. Table of Contents...List of Tables.....List of Figures....Introduction....Procedure.... Experimental Program...Experimental Data.Results....Discussion....Conclusions.....Recommendations.....Bibliography...Appendices.....On Time Submission.....Group Activities Evaluation Forms (To be handed in confidentially) ..

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Discussions Interpretation of the results Comparison with known or published standards

Conclusions Validity of results Applicability to other fields Universal or specific conclusion

Recommendations Additional experiments Applicability and correlation to other engineering problems Improvement suggestions to reduce error and uncertainty Other techniques

Assessment completed by Team Members: (Print)

Name: Emily DiMartini

Name: Dan Quiroga

Name: Monica Sharobeam

Name: Brandon Simon

AbstractIn this experiment, a beam was loaded in increments and three different strain measurements were taken at different points on the beam. This was done to verify the linearity of strain along the axis of the beam. These strain measurements were subsequently converted to shear stresses, and further to bending moments along the beam. The shear stresses and moments were graphed along the length of the beam. The experimental measurements confirmed that the shear stress was constant along the length of the beam and the bending moment varied linearly, as expected.

ContentsAbstractiContentsiiList of TablesiiiList of FiguresiiiIntroduction1Procedure1Experimental Program2Experimental Data3Results4Discussion9Recommendations10Conclusion10Bibliography11AppendixA

List of TablesTable 1. Collected beam data4Table 2. Three strains associated with each individual loading weight4Table 3. Increasing Load Shear Force5Table 4. Decreasing Load Shear Force5Table 5. Calculating Shear Force from Slope of Best Fit Line7Table 6. Bending Moment8Table 7. The equation of best fit with the identified shear8

List of FiguresFigure 1. Top view of the beam3Figure 2. Front view of the beam3Figure 3. Strain and Distance with Increasing Load6Figure 4. Strain and Distance with Decreasing Load6Figure 5. Free Body Diagram, Shear and Moment Diagrams7Figure 6. Bending Moment in Relation to Distance and Load8Figure 7. Schematic of strain indicator setupA

11

IntroductionThis laboratory experiment will introduce students to measurements of axial strain using three strain gages. Loads were applied to the aluminum beam in cantilever form, causing deformation in the direction perpendicular to the axis collinear with the beam. The purpose of the experiment is to determine the shear force and load from strain measurements, to verify the linearity of strain along the beam axis, and to confirm the shear force and moment relationships by comparing two different stress determinations. The calculated stress and bending moment values correlate to a relationship based off the integration of the shear stress over the length of the bar.

ProcedureThe first part of the experiment is to set up the strain indicator. In order to do this, turn the strain indicator on 15 minutes before use, and then install the beam in the back of the frame. Record the cross section dimensions, the lengths from the strain gages to the point of application of the load, and the measurement group, gage type, resolution, gage factor and correction K specific for the specimen. Attach the wire connections according to the diagram in Appendix A and connect the strain indicator to the three channels. Then set the gauge factor in the indicator and zero the strain.The second part of the lab involved applying loads and taking measurements. Apply loads with an S hook in increments of 500, 700, 900, 1000, and 1100 and then unload the beam in those increments. Be careful not to exceed these increments; this risks permanent damage to the beam. Record the resulting strain in the three channels. Then draw a figure of the test setup, a free body diagram, a shear diagram and a moment diagram. Experimental ProgramThe materials and conditions required for this laboratory experiment are:Tools:1. Cantilever flexure frame1. High-strength aluminum alloy beam, [with approximated dimensions of: 1/4 x 1 x 12-1/2 ( 6mm x 25mm x 320mm)]1. Digital Calipers - capable of measuring 0.001 in1. Accurate 12-inch scale ( 1/64 in)1. Vishay P3 Strain Indicator1. 500g mass set (with increments of 100g)Variables: Independent Variables: applied load Dependent Variables: Axial strains 1, 2, and 3; experimental moment, experimental shear Control Variables: type of bar used Extraneous Variables: Temperature, Residual voltage

The procedure consists of measuring the three axial strain values, each a specified distance from the load. The shear stress and moment were calculated from the given youngs modulus and measured strain values.Sources of error and limitations of this experiment include: hysteresis Room temperature (thermal strain) Clamp not being tightly secured on prismatic barThese errors and extraneous variables can be considered negligible. The effect of hysteresis of the loading process was counteracted with unloading, giving a set of data that includes over and undervalued points to account for the beam not perfectly following Hookes law. The variation in temperature would account for some amount of strain, however, because of thermal strains linearity, it will affect all the strain gauges the same. Temperature variation within the bar would cause the most notable extraneous error. Vibration through the system can cause a residual voltage, another extraneous error. Residual voltage caused by a change in voltage would be a minimal systematic error. The clamp not having the appropriate grip on the bar would cause minimal systematic error.Experimental DataFigures 1 and 2 below illustrate a diagram of the aluminum alloy bar used in the experiment.L2

L3L1b

Figure 1. Top view of the beamL2L2L1tP

Figure 2. Front view of the beamThe measurements t, b, L1, L2, and L3 were measured and can be found in Table 1. Along with the dimensions are the specimen data used for reproducability of the experiment. These values were later used in further calculations.

Table 1. Collected beam dataValue

b25.22 mm

T6.67 mm

L1107.95 mm

L276.2 mm

L376.2 mm

Measurement GroupB-105

Gauge Type125AD

Resolution120.0

Gauge Factor2.1 5%

Correction K+1

At point P weights were loaded and then unloaded, and the associated three axial strains were measured and can be found in Table 2.Table 2. Three strains associated with each individual loading weightmass (g)123

5001097546

70015310365

90019613184

100021814593

1100240159102

100021814393

90019612784

7001539565

5001096247

01-151

ResultsThe measured strain measurements were used to calculate and (All values of strain are measured in micro strain.) These values were used to calculate the shear force using the equation below. (1)E= 68.9 GPab=width of the section (m)t=thickness of the beam (m) = change in strain (no units)= change in distance (m)

The tables show the values for and as well as the calculated shear force (V) when the load was increasing (Table 1) and also for when it was decreasing (Table 2).

Table 3. Increasing Load Shear ForceMass (g) (m) (N) (m) (N)(N)

500-34-0.7624.61-29-0.7623.934.27

700-50-0.7626.78-38-0.7625.155.97

900-65-0.7628.82-47-0.7626.377.59

1000-73-0.7629.90-52-0.7627.058.47

1100-81-0.76210.98-50-0.7627.739.35

Table 4. Decreasing Load Shear ForceMass (g) (m) (N) (m) (N)(N)

1100-81-0.76210.98-57-0.7627.739.35

1000-75-0.76210.17-50-0.7626.788.48

900-69-0.7629.35-43-0.7625.837.59

700-58-0.7627.86-30-0.7624.075.97

500-47-0.7626.37-15-0.7622.034.20

The two graphs below show the relationship between the values of strain and distance. Distance was measured in meters and strain was measured in micro strain.

Figure 3. Strain and Distance with Increasing Load

Figure 4. Strain and Distance with Decreasing Load

Using Figures 1 and 2, as well as a best fit line, the shear force was calculated. The slope of the line of best fit is equal to the ratio of and By multiplying this ratio (slope) to the shear force was determined.

Table 5. Calculating Shear Force from Slope of Best Fit LineApplied Load(N)Best-Fit Line EquationSlope from Best Fit Line ( (m3Pa)V (N)

4.90y = -4.13E-4x + 1.23E-64.13E-410,3354.26

6.86y = -5.77E-4x + 1.71E-55.77E-410,3355.96

8.83y = -7.35E-4x + 2.19E-57.35E-410,3357.59

9.81y = -8.20E-4x + 2.31E-58.20E-410,3358.47

10.79 y = -9.05E-4x + 2.68E-59.05E-410,3359.35

A Free Body Diagram of the beam was drawn. From this the sum of the forces was indicated as well as the sum of the moments in the beam. From that, the shear and moment diagrams were determined.

Figure 5. Free Body Diagram, Shear and Moment DiagramsThe bending moment was calculated using the equation determined from the moment diagram.

After this, it was graphed. Moment is measured in N*m. (2)Table 6. Bending MomentLoad (N):4.905 6.86 8.829.8110.79

M()-1.27-1.78-2.29-2.54-2.79

M()-0.897-1.26-1.61-1.79-1.97

M()-0.523-0.733-0.942-1.04-1.15

Figure 6. Bending Moment in Relation to Distance and LoadBest fit lines were generated from the bending moment graph (Figure 4.) The slope of these lines is equal to the load as well as the shear force, as shown in the shear diagram, Figure 3.

Table 7. The equation of best fit with the identified shearApplied Load (N)Equation of line of best fitSlopeShear Force (N)

4.90 y = 4.90x - 1.454.904.90

6.86 y = 6.86x - 2.026.866.86

8.82y = 8.82x - 2.608.828.82

9.81y = 9.81x - 2.899.819.81

10.79y = 10.79x - 3.1810.7910.79

DiscussionThe objectives of this lab were teach us how to determine shear force and the load from strain measurments, verify linearity of strain along the beam axis, and confirm shear force and moment relationships by comparing two difference stress determinations. The lab also allowed us to better analyze the relationship between shear and moment. Using strain values as well as distances we were able to calculate the shear force (N) for each of the loads using Equation 1. Table 1 shows this for increased loading and Table 2 shows this for decreased loading. In addition to calculating the shear force, we also graphed the three strain values for each load over the distance. This was done because the strain values are all linearly related to one another for each load. After this, best fit lines were produced. The slope of these lines is related to Equation 1 because it is equal to . With this we multiplied the slopes of each line by 10,335 (m3Pa), which comes from calculating with the appropriate values. This answer was the shear force. The values for the shear force from the best fit lines all equaled the average shear force calculated in Tables 1 and 2. This proved the data to be accurate. In addition to this, we drew a free body diagram, summed the forces in the x and y direction, as well as summing the moment equal to zero. From this, we created a shear diagram where the load was equal to the shear force. Along with this and boundary conditions for the shear force, we created a moment diagram. The equation for the moment diagram can be seen in equation 2. To further investigate the relation between bending moment and shear force, we generated best fit lines for Figure 4. The slopes of the best fit lines were equal to the shear forces as well as the corresponding loads, which we concluded would be the case from the shear force diagram in Figure 3. This also proved that this experiment was successful. Errors in this lab mainly came from extraneous variables. For instance, a common problem is that the table is never completely stable. There are many groups working at the same time, and it is hard for the table to remain still. These vibrations from movement cause error in strain values, which leads to miscalculations. Lastly, another error in this lab comes from the loads touching the table. This would alter the amount of load on the beam. RecommendationsIn this experiment, axial strain was measured and used to find the shear force and bending moment for the beam. This confirmed that strain is linear along the beam, and that the bending moment is the integral of the shear force. This could be applied in further experimentation. Different beams could be tested to see how the bending and shear moment graphs change. Further, by applying multiple loads at different points in the beam, one could examine how this affects the linearity of strain and the subsequent stress and moment graphs. A uniformly distributed load would make a more complex and interesting stress and moment graph.Overall the experiment and procedure were straightforward and led to accurate results. One issue was can when unloading weight increments, the strain measurements for channel two were off. This was accounted for with random loading, but multiple trails would be beneficial. The strain gauges accurately measure strain and yielded successful results.ConclusionThe goal of this experiment was to determine the relationship between shear stress and moment. Shear stress remained constant while the moment changed linearly. Procedures completed in this lab to determine shear stress and moment on a beam apply to many concepts such as diving board or large scale engineering projects, such as steel beam design for expedited analysis of loads. Understanding how structures resist forces is important when designing. The experiment was done successfully because the shear stress and moment diagrams followed the theoretical results expected. The relationship between shear stress and moment was understood through graphs and shear and moment diagrams.Bibliography1. Amirault, S. B. Shear and Moment Diagram. Digital image. Shear and Moment Diagrams. S.B.A Invent, n.d. Web. 1 Apr. 2015.

Appendix

Figure 7. Schematic of strain indicator setupA